When you open the first small key within City in the Sky, a cutscene trigger is added at the end of the bridge that you came in from, and your savewarp point is changed to start you off next to the small key chest until the cutscene is triggered. This savewarp spot is incredibly convenient because you return to this room immediately after obtaining the Double Clawshots, which saves some backtracking.
After obtaining the first small key, head back outside and cross the bridge. Target the last peahat and sidehop onto the banister on the right on GCN (it will be the last broken chunk before the staircase). Jump to the next banister then head back into the dungeon through the broken grate window, staying clear of the door at the end of the bridge.
Run straight for the center of the platform on the other side of the room, and perform a spin attack. This will knock the two Dynalfos off of the platform. Quickly target the vines with the clawshot, and grapple to them before the cutscene begins.
Just roll through the first two fans. Then to get past the last fan, equip the iron boots and quickly sidehop across the falling blocks.
After the battle has begun, walk forward onto the yellow block until you are about a foot from the edge. Turn around and carefully aim the clawshot at the medallion behind the gates to get to the chest without beating the aerlfos. To get out, simply savewarp.
Start by hitting the Aeralfos with either the Bow & Arrow or Boomerang to make it pull out his shield, then use the Clawshot to pull him down to the ground. Take out the Ball & Chain, swing it into him four times, and release it on the fifth swing. The Aeralfos should block the last attack and counter it, however when the ball returns it will hit him one last time. For it's second phase, grab him towards you with the Clawshot and swing the Ball & Chain like before to defeat him.
Simply use a few Long Jump Attacks to cover the gaps without waiting for the updrafts to disappear.
Since boomerang Long Jumps can't be done on the Wii version, instead throw the Gale Boomerang at the Keese conveniently located near the gap. As the boomerang starts to return towards you, target the Keese in order to do a Long Jump. Repeat this process to cross the final gaps.