Jirutagumac Shrine can be found in the Lanayru Great Spring Sky, west of the Water Temple. There you will learn how to use the Wing, a Zonai Device.
Shrine Location
Jirutagumac Shrine is west of the Water Temple, at 2916, 0533, 0951.
A Flying Device
There are Wings up ahead, and a rail that you can put them in. Wings will automatically appear on the rails, you can either hop on before it slides away, or use Recall to make one come back to you so you can hop on, or you can use Ultrahand to put a Wing on the rail.
Stand on the footprint symbol in the middle of the Wing to keep it flying straight ahead, or carefully walk onto the left or right wing to make it turn in that direction. You can step on the tail of the Wing to make it go upward, or step on the nose of the Wing to make it go downward. Be careful not to fall off. If you stand on the tail of the Wing, it will lose speed, and may begin to fall straight down.
Use a Wing to go to the area down below on the right, where there is a Fan and a Wing. There is a treasure chest on a platform sticking out of the wall. Take the Wing and place it like a bridge across the gap under that platform, then use Ascend to reach the platform. Open the chest to get a Large Zonaite.
Go to the other side of the area, where Carts are falling down from the ramp above. Use Recall on a Cart to stop it, then use Ultrahand to pull it to you.
Then attach a Wing to the top of the Cart, and attach a Fan to the top of the Wing's tail. Take the Wing to the upper area (where you found the Fan) and point it at the final area of the shrine. Stand on the Wing and hit it to start the Fan, then walk around on the Wing as needed to steer it to the final area.
Check on the dragon ring to get a Light of Blessing and complete the shrine.