Instructions
Obtain the third fused shadow from the Lakebed Temple and return back to Hyrule Castle. After completing the Lakebed Temple, Midna will be attacked and fall ill. You must return to Hyrule Castle so Princess Zelda can heal Midna and also grant you several new abilities, including transforming into a wolf whenever you want and warping through the game world.
Approach the Bridge of Eldin from the north end to trigger a battle with shadow monsters. They must be defeated in order to gain control of the shadow gate that is hovering above the bridge. Once they are defeated, you can now warp to the gate at any time.
Use the large cannon at Lake Hylia to enter Gerudo Desert. To gain access to the cannon, you will first need to meet with Telma and her resistance in her bar, located in Castle Town. Telma will send you to the lake to meet with an ally. You'll find him atop the lookout tower on the west side of the lake. He will give you a note that will allow you to go to Gerudo Desert via the cannon.
Head to the southeast corner of the desert and warp the missing bridge piece back to Eldin. The missing piece of the bridge is located atop a small mesa in the desert. As you get near it, Midna will begin talking about the strange block. Use your warp ability near the stone to teleport to the Bridge of Eldin. If you're close enough to the stone, it will travel with you, falling back into place and repairing the bridge permanently. The top of the desert mesa will be clear as well, giving you access to the Cave of Agony.