Miryotanog Shrine is west of Gerudo Town in the Gerudo Desert.
Shrine Location
Miryotanog Shrine is located at -4679, -3086, 0054, which is west of Gerudo Town in the Gerudo Desert.
Proving Grounds: Lure
Take the weapons from the rack near the entrance. If you stand in the laser beams, three large metal orbs will roll down the ramp, and you will have to get out of the way to avoid getting hit, or catch one of them with Ultrahand, or use Recall on one. To the side of the ramp, there are two wooden crates, with a fire fruit tree behind it.
The metal orbs themselves will trip the laser beams, making more of them fall. (If this doesn't happen, stop one of the orbs, then drop it into the pit a few moments later.) This can be helpful if you want to lure the Constructs into the path of the orbs.
When there is only one Construct left, on the other side of the water, use Ascend to get onto the raised walkway above the area where the orbs roll down the ramp. Take one of the orbs and drop it on the ramp on the side of the room with the final Construct. If you aim it correctly, it will hit the stone platform that the Construct is on. (If not, use Recall to bring it back and try again.) You can then use an object to trip the laser near where the Construct fell, and the flame emitter in the wall will burn the Construct.
After you defeat all the Constructs, the gate opens. Go through and open the chest to get a Captain IV Blade or similar. Then check on the dragon ring to get a Light of Blessing and complete the shrine.