After you solve the riddle of the mural with the pillars of light, the Lightning Temple emerges from the sands.

Find a Treasure Chest
In the south corner of the first room, you find a Korok-Leaf Guster. You can use this to blow away the piles of sand in the room. This will reveal a treasure chest.
Find a Switch
Blow away the southwest pile of sand to find a switch. Step on the switch to open the door.
Find a Second Treasure Chest
Go southwest down the stairs. You will cross a bridge, then a second bridge. Before you go through the door at the end of the second bridge, float down below this bridge, defeat the Gibdo that you find there, then open the treasure chest. Then use Ascend to get back on top of the bridge, and go southwest.
Find a Third Treasure Chest
There are loose bricks in a hole in the southwest wall, so use Ultrahand to move them out of the way, then go into the room beyond and defeat the Gibdo there, then open the treasure chest. Before you go, remove one of the stone slabs to find a Gerudo Claymore.
Find a Hidden Room
Exit the room and carefully go northwest, but beware that the stone in the middle will fall. You can move it out of the way with Ultrahand. Go down to the room below and defeat the Electric Keese, then move a stone slab from the niche in the wall, and you will find a Gerudo Scimitar. Use Ascend to go back up.
Get Past the Fire
There are flames in the room ahead, and the large block that led to the hidden room is too large to go through the door. But there are smaller stone blocks in the previous room (the ones that were blocking the entrance to a tunnel with a treasure chest in it) so go southeast and get them, rotating them as needed to make them fit through the doors, and place them in front of the jets of fire.
In the room beyond, step on the switch to stop the flames and open the bars. Go through.
Flaming boulders are rolling from the hallway ahead. You can use Recall to make a boulder go in reverse. Quickly run along the hallway behind the time-reversed boulder and go through the opening at the end before a new boulder can appear.
Step on the switch in the room beyond, and this will stop the fireballs so Riju can join you. This also opens the bars to the next area, so go through.
Destroy a Hive
There is a Gibdo in the hallway ahead, so have Riju create lightning, and shoot the Gibdo. Then go ahead to the next room. In the east corner of the room, there are stairs leading down into the pit. Go down these stairs and move away from the Gibdo Hive to make it start glowing, then use Riju's lightning to destroy this hive and defeat any Gibdos that popped out of it.
You can move the stone slabs here to find a Gerudo Bow.
Before you go back up, go southwest through an opening in the wall. You will find a spinning object. Use Recall on it so that its spikes won't be moving toward you, and let the spinning object take you up to the ledge behind it. Open the chest to get a Mighty Zonaite Shield.
Then go back to the room where you destroyed the Hive. Go up the southeast stairs.
Room of Hopeful Light
You reach the Room of Hopeful Light. Defeat the Construct, then in the southern corner of the room, on the west side, move the stones out of the way to find a hidden room. The treasure chest is sitting on a stone that will fall when you stand on it. Open the chest, then in the area down below, where the stone fell, you can place the chest on top of the flames to reach the area behind them, where you find a Gerudo Spear.
Take the two stones that you moved out of the way of the corner, and attach them so that they will be as tall as possible when placed on the floor. Then you can use Ultrahand to pull up the mirror that is sticking partway out of the sand. Attach the mirror to the top of the stacked stones, and move this into the beam of light. Place it so that the light is glowing at the jewel on the wall above the locked northeast door. Go through the door that opened.
Room of Ascension
You will reach the Room of Ascension. Make sure Riju follows you into the room. There will be a cutscene when she arrives. Then check on the glowing terminal in the stone ring to create a teleport back to this room (it won't work until after the cutscene with Riju).
In the southern corner of the room, remove the stone block from the floor to reveal a wind spout. You don't need this right now, but it will be helpful later.
Activate a Battery
In the east corner of the room, look in the wall to find two more stone blocks. Pull them out, then go through the opening that they were blocking. Here you will find one of the batteries. Have Riju create a lightning circle, then hit the battery when it is within the circle.
Room of Light and Flame
Go to the southern corner of the Room of Ascension and go northeast up the stairs. As soon as you are at the top of the flight of stairs, you can go southeast into a room. This is the Room of Light and Flame. Defeat the two Constructs. One of them is holding a Mirror Shield.
You can't reach the battery from here, so exit the room. Go northwest up the stairs, and as you go, notice that there is a jewel in the northwest wall, but you currently don't have a way to light it up, so just keep it in mind for later.
Keep going up until you are on 4F.
Activate a Second Battery
Look at the map of 4F, and look at the north corner for a hallway that is going northeast. Go around the walkway to the south corner of 4F and use the wind spout that you uncovered on the lower floor to glide north and enter that hallway.
Defeat the Construct in this room, then go through the opening in the rotating stone in the wall. Go through the opening in the next rotating stone, and in the room at the end, pull the stone blocks out of the wall. This reveals a bright light. Go back through the rotating stones. Since the rotating stones block the path of the light, you will have to use Recall on one of them until the time-reversed one is in the same position as the other, then press L to turn off Recall so that the two rotating stones are aligned and the light can come out.
The bars will open and the statue will move. Wait for the rotating stones to get out of the way of the beam of light again, and this time the light will be able to reach the jewel on the statue across the room. The statue will move, revealing a switch beneath it. Go to that switch and step on it.
Go into the room beyond, to find some spikes closing and opening. Place one of the stone blocks between the spikes to stop them from closing, then go past the spikes. Then step on the switch in the east corner of the room to stop the spikes so Riju can join you.
In the next part of the room, pull the stone blocks out of the wall and place one beneath the spikes that are going up and down, then go past it and defeat the two Constructs.
Find the hole in the northeast part of the floor and drop down it. The room is spinning, and there are Zonai Spikes here. Spikes can be pushed into solid materials with Ultrahand, and when you let go, they will stay stuck in the wall. So push a Spike into the solid wall (not into the spinning wall), and it will stop the spinning room. Then look for the two lamps side-by-side in the southwest part of this room. You need to stop the room so there is an opening between those two lamps, because there is a jewel on the non-moving wall there. If you stopped the room but did not reveal the jewel, put the second spike into a solid wall, then remove the first to let the room rotate slightly, and keep switching the spikes like this until you have revealed the jewel in the wall. Once you have stopped the room and revealed the jewel, place a mirror on the light and point it at the jewel. This will open the trap door above.
Use Ascend to go to the room above, and pull the Mirror out of the sand in the southeast part of the room, and set the Mirror on the walkway above the beam of light. Point it down the hallway. If the light is blocked by the stone that you used to block the spikes, there are Zonai Spikes in the corner of this room that you can pick up with Ultrahand. Push one into the wall beneath the spikes that were moving up and down until you put the stone block beneath them, then go northwest to the end of the hallway and remove the stone block that you put beneath the spikes. This will allow the light to shine on the jewel. This will open the gate, revealing a battery.
If Riju is not with you, step on the switch in the east corner of the room to stop the spikes, and remove the stone block from between the spikes if there is one. Then backtrack until Riju joins you, then talk to her to create a lightning circle, and go toward the battery that you revealed so that she will come with you. Shoot the battery with an arrow to activate it.
Use the map to look at floor 1F, and teleport back to the Room of Ascension. Use the wind spout to go to 5F. Go to the middle of the southwest wall and climb up into the tunnel there to find the Room of Natural Light.
Room of Natural Light
If there is already a stone block with a mirror on it that is shining the light through the opening in the northeast wall, skip this paragraph. In the western corner of the room, there are mirrors in a niche in the wall. Move the stone block into the beam of light so that it is standing as tall as possible. Then attach a Mirror to the top of it and turn it so that the beam of light is shining through the tunnel in the northeast wall.
Exit this room via the northeast tunnel, then use Ultrahand to move the statue left or right on its rail until its mirror reflects the beam of light. Go to where the beam of light is shining, and use Ultrahand to move the nearby statue so that it reflects the light. Be sure to move it all the way to the end of its rail so it is shining the light at the jewel on the statue down below.
Go down to where the beam of light is shining, and step on the switch that was revealed there. You will reach the Room of Offered Light.
Room of Offered Light
There are many Constructs in this room, and Gibdos will start appearing from the Hive. Defeat any Constructs that get in your way, and have Riju create lightning, then wait for the Hive to start glowing brightly, and shoot it. Defeat the remaining enemies.
Go to the balloons near the northeast wall, and hit the balloon that is standing on a metal plate. This will break the urn in the balloon. Attach this balloon to the metal plate, then move it into the beam of light. Attach a mirror to the northwest edge of the balloon platform (the side that points toward the battery behind the gate), with the mirror facing outward to shine a beam of light northwest. Take the torch from the holder in this room if you need it, then light it and stand on the metal plate to make the balloon float up. There is a jewel above the gate to the battery, and the mirror will shine on it if it is aimed correctly.
The gate to the third battery will open, so talk to Riju and shoot the battery when it's in the lightning circle.
Activate a Fourth Battery
Use the map to look at 1F, and teleport to the Room of Ascension. Use the wind spout to float up to floor 5F, where there are four statues on rails. Go to the southwest wall, where light is coming out of the tunnel in the wall. Move the statue out of the way of this light if it is in the way, so that the beam of light goes straight across the room. Then go to the northeast wall, use a guster or fan to blow the sand away (or if you have completed the Wind Temple, you can use the item you got after completing that temple, and blow a gust at the sand). After the statue is out of the sand, move the statue into the path of the beam of light. Then go to the northwest statue and move it until it shines its light on a jewel in a statue up above.
Go to the southeast wall and climb on top of the base of the statue there, then use Ascend from there to reach the statue above that, then use Ascend again to reach the statue that just moved, and step on the switch to reveal a tunnel. Go down the tunnel and you will find many flames in the hole below. Carefully glide down while avoiding the flames. You will find the last battery in the room where you land. Step on the switch to open the door, kill the Constructs (if any), and talk to Riju to activate her lightning to activate this last battery.
Activate the Room of Ascension
Go north to the tunnel that leads to the Room of Ascension, and go down to the middle of the room. Check on the glowing terminal in the stone ring and there will be a cutscene.
Room of Glorious Light
Talk to Riju, and shoot the glowing Hive above where you can see something shining behind the wall. There will be a cutscene. Then you must battle Queen Gibdo.