I just finished the game this morning, so that part is still pretty fresh in my mind.
Since you have the hanging cage already out, go ahead and turn the crank again that opened the floor (to close it.)
Now, use your rope-bow to pull the cage back and break the wall open. It will reveal a wall you can climb.
Then lower the hanging cage, so that is hangs at the level of the floor.
Now what you need to do after that is climb to the highest platform in the room.
Use your rope-bow again to pull the hanging cage back. When it swings towards you, quickly jump on top of it, ride it towards the wall, and jump.