Use the grappling hook and swing to the other side.
Do a backwards jump when you've reached the end of that wall mount.
The rod will swing. And you can jump into the enclosure and push the button.
Another rod will turn. Last one. I promise.
Get down, and back to the no. 3 mattress.
Climb up, and again to the right side.
This time do a backwards jump onto a rod, and then start rotating. Jump and catch the second rod.
Start swinging again and grab onto that pole.
Jump on the second one.
And then on the first raised pole.
Continue jumping until you've reached the platform or you're dead. Whichever comes first.
Grab the Wrench. Congrats. You're done with this room. Not very much left now.
Go back down and back to the living room.