You find a way into the Gerudo Town Emergency Shelter, where you learn that Riju is training in the northern ruins.
This quest begins when you enter the Gerudo Shelter in Gerudo Town. If the guard won't let you in, you can go behind the throne and use Ultrahand to move the Sand Seal statue and reveal a hole that leads down to the shelter.
If you go to the northeast exit of town, then go directly north, you will reach the ruins. Just keep in mind that your map and compass won't work in the sand shroud, so try not to change direction.
When you are close, you will see lightning strikes. Go toward these to find Riju. Approach her, and there will be a cutscene.
Help Riju Train
Get out from behind Riju, and wait for her circle of lightning to expand so that the dummy is within it. Then shoot an arrow at the target on the dummy.
Next, wait for Riju's circle of lightning to expand to reach the rock in the middle of the three dummies, then shoot an arrow at that rock.
Go to Kara Kara Bazaar
Before you leave the ruins, open the treasure chest behind the nearby stone column.
Then teleport to Mayatat Shrine. There will be a cutscene.
When the battle starts, use the skills that you learned with Riju at the ruins. Shoot arrows so that they land in Riju's circle of lightning, near some Gibdos. The locations of Gibdos will be marked in red on the map.
After you have defeated them, Riju will notice that they are coming out of the glowing mushroom-like object. Wait for the mushroom to be inside of Riju's circle of lightning, then shoot the glowing core of the mushroom. After that, shoot any remaining Gibdos, and there will be a cutscene.
Return to Gerudo Town
Teleport back to Soryotanog Shrine, then go down to the quest destination to find Riju. Talk to her.
After the cutscene, enter the throne room and talk to Buliara. She says to talk to Captain Teake to discuss strategy, and talk to Padda to get supplies.
Talk to Padda to ask her to set up a barricade at one of the gates. She has supplies to help you in the battle, so keep this location in mind in case you run out of arrows or bows, or other things.
Talk to Teake to deploy troops at the gates that you choose. The battle can be won regardless of how you deploy the troops and barricade.
After the barricade is in place and you have deployed soldiers to all three gates, talk to Riju and tell her that the preparations are complete. After a cutscene, the battle will begin.
Battle the Gibdo Horde
It is recommended to climb up to the top of a wall so you can quickly go to whichever gate needs the most help.
Check your map to see where the enemies are. Your top priority is to defeat any enemies that reach Riju and are attacking her directly. If there aren't any, then defeat any enemies that have gotten past the gates and are going toward Riju. Otherwise, your priority is to destroy the Gibdo Hives as soon as possible. There is a Hive at each of the three gates and are marked with a large red dot. The Hives can only be destroyed when they are glowing brightly, so stay near one (if there aren't enemies attacking or approaching Riju) and shoot it with an arrow as soon as it is glowing brightly and is within Riju's lightning circle.
Eventually, flying enemies will start to attack, and they can quickly reach Riju, so defeat them as soon as you can.
Keep battling, protecting Riju if any enemies get close to her, and destroy the hives as soon as you can. After the hives are gone, defeat any remaining Gibdos to end the battle. If there don't seem to be any enemies, be sure to visit each gate in case there are any there.
Collect the Spoils
Before you proceed, you can go around town collecting all of the monster parts. If you have unlocked Sensor+, you can take a picture of a Gibdo Bone and set the Sensor+ to search for it. (You can't add Gibdo Wings to the Hyrule Compendium.) Padda has some in the courtyard southeast of the throne room, as well.
Examine the Mural
Go into the emergency shelter (talk to the guard to open the door) and go to the quest destination to find Riju. Talk to her and there will be a cutscene.
Go to the First Pillar
Go back toward the main room of the shelter. Near the door to the room where you talked to Riju, there is a part of the ceiling that is higher than the rest. You can use Ascend here to emerge behind the throne. (Or you can just walk to the throne from here.) When you are behind the throne, go southwest past the sand seal statue. You will see one of the pillars that was depicted on the mural. Go to it, but beware of Gibdos on the way. When you reach the pillar, use Ascend to reach the upper platform. You will find some broken stones covering a glowing part of the platform in the center. Use a Bomb Flower to destroy these stones and light up the pillar.
Go to the Second Pillar
Then follow the light from this pillar. There are some wind spouts on the way, but there are also Gibdos along the way, so be careful.
At the pillar, you discover that the light lens is too low for the beam of light. However, there is a stone structure nearby. Go to that structure and use Ascend to reach the top. Then push the wheel to make the pillar go higher, until the light hits the lens.
Go to the Third Pillar
Even though the light is connecting the pillars now, you still need to go to the pillar that you haven't been to yet, because the final beam isn't pointing exactly at the first pillar. From the second pillar, follow the beam of light that goes to the right. You will find a wind spout near this third pillar. Float up in it and land on the pillar, then break the cracked stone in the middle, and remove the Hover Stones from the pit in the center. If you use Ascend to reach the top of the pillar, you find a round device there, but there is no way to use it in its current state. So go back down to where you took the Hover Stones out of the pit. Attach at least one of the four wooden pieces to one of the Hover Stones, then hit the Hover Stone to activate it. You can now use Ultrahand to move the activated Hover Stone up to the raised walkway. It won't be able to go all the way up, but since you activated it, the Hover Stone will float wherever you put it. So use Ascend to get onto the raised walkway, and use Ultrahand to move the Hover Stone near the top of the pillar. You won't be able to make it go all the way to the top, so float back down to the lower part of the pillar and use Ascend to go to the top, where the round device is. Then use Ultrahand to move the Hover Stone onto the top of the pillar, and hit it to deactivate it.
Detach the wooden piece or pieces from the Hover Stone, then attach them to the square sides of the round object so that they stick straight out. Then push one of the wooden pieces to spin the round object. This will change the angle of the beam of light shining from this tower, so keep an eye on that beam of light to see which way it is going, and make sure that it is moving toward and not away from the other pillar. When the beam of light hits the first pillar, there will be cutscene that shows the lightning stone appear in the center of the triangle.
Get the Lightning Stone
Then you can float down and go to the center of the triangle (when the map stops working, you can go uphill to find it), and you will find the Lightning Stone. Approach it and there will be a cutscene. Afterward, go up to Riju and talk to her. When she makes a circle of lightning appear, shoot the Lightning Stone's pedestal with an arrow. There will be a cutscene.
Approach the Building
Approach the building that appeared. This is the Lightning Temple. Talk to Riju to have her create a circle of lightning, then shoot the large glowing object in the front of the temple. There will be a cutscene, and you will have to battle Queen Gibdo.