How do I open the shaking door on the 20th floor?

You can't open it in its present state.

Or perhaps more accurately, you can't open it in *your* present state.

Look for something that might help alter your state.

It was also described in the journal that came with the game.

The only room you can access so far on this floor contains a chair of "a
thousand fingers."

Maybe you should try sitting in it.

But you don't know how to operate it.

There's a barrel just next to the chair with a crank in it.  If you turn the
crank, someone claiming to be Dr. Sartorius will come out and offer to give
you a treatment in the chair.