The Keh Namut shrine contains the Cryonis trial, where you get a new rune for the Sheikah Slate.
How to Reach the Western Shrine
The next shrine is in the southwest mountains of the Great Plateau. From the Owa Daim Shrine (the southern shrine) you can go northwest up a slope to reach these mountains. Another way to get there is to go up the slope behind the Temple of Time that leads upward into the mountains. However, the mountains are cold, so you will need protection from the low temperature.
How to Protect Yourself from Cold
If you didn't get the Warm Doublet from the old man, read the Stasis Trial page for more information.
If you don't have the Warm Doublet and don't feel like getting it, one way to protect yourself in cold temperatures is to light a torch and carry it. The problem with this is that if you put the torch away (which you automatically do if you run), the torch will be extinguished, and there might not be a fire nearby to light it again.
Another way to protect yourself is to cook food that gives a cold-resistance bonus. You will find Spicy Peppers near the entrance to the mountain area. You can cook these by themselves or combine them with meat and fish and other ingredients to create food that provides temporary protection from the cold. Be sure to cook plenty of cold-protecting foods before you try to reach the shrine.
Go to the Shrine
After you have some protection from the cold, go into the mountains as explained above in the How to Reach the Western Shrine section.
The River of the Dead is in the southwest part of Great Plateau, and the shrine is west of there. Don't try to swim in the River of the Dead, because the water is freezing cold, and you will freeze to death if you get in, regardless of any protection you might have against cold.
If you walk along the path that runs along the north edge of the River of the Dead, you will find a ruined cottage with a dead statue in it. Search the statue to get an ancient item, then take the bundles of wood. Also, use Magnesis on the water near here to get a treasure chest that contains Amber.
Continue going west along the north edge of the River of the Dead. Ice Chuchus might attack you as you go along. Be sure to get out of the way after hitting them, because when they die, they will emit a burst of freezing air that will temporarily freeze you if you are too close.
Eventually, you will go past a broken stone wall. Use Magnesis on the metal door that is leaning nearby. Put it onto the wooden poles in the water. If possible, make the door sit flat on the poles. Then walk across the bridge to go past the water. From there, go up the nearby slope and make your way up the path that spirals up the hill. You will encounter some Bokoblins on the way, unless you climb up the side of the hill just before the top.
Get the Cryonis Rune
Inside the shrine, check on the pedestal to get the Cryonis Rune added to your Sheikah Slate.
How to Use the Cryonis Rune
Switch to the Cryonis Rune and press L to start it. Aim the reticle at water and press A to create a frozen platform of ice. You can climb to the top of the ice platform. You can also aim the reticle at an ice block and press A to shatter it. You can also aim the reticle at some waterfalls to create sideways platforms. You can also press down on the directional pad to switch between aiming underneath you and aiming far away. This is useful for creating a platform if there is shallow water that you can stand in.
Reach Keh Namut
Stand near the ledge and use Cryonis (press down on the directional pad to aim underneath you) to go up to the ledge. In the next area, press down on the directional pad to aim far away (if you aren't already), then create an ice column under the bars. This will lift the bars.
There is a Guardian in the next area. One way to fight this type of guardian is to raise your shield, then if you press A (not Y) right before the Guardian's beam hits you, you will do a Perfect Guard, knocking the beam back at the Guardian to hit it. You can also make pillars of ice and hide behind them. Or you can just try to shoot the Guardian in the eye before it charges up its laser.
Near where you fought the Guardian, there is a treasure chest on a ledge. Use Cryonis to reach the ledge and open the chest, which contains a Traveler's Spear.
In the next area, there is a see-saw. Use Cryonis to make a pillar of ice under one side of the see-saw. Walk up the see-saw to reach the upper area. Then go up the stairs to check on Keh Namut and get a Shrine Orb.