You need at least a 2x3 and a 2x4 would be much better. I think what was meant by corner cage is an L-shaped cage, which is fine.
Here's what I would do:
Go through your room and throw away or give away everything you don't really want.
Organize what remains so you have as much space as possible. Move stuff on shelves and use as much vertical space as you can.
If there is any furniture you don't really use, move it to another room.
Measure how long each side of your room is and draw it on graph paper to scale.
Measure each thing that you have to keep in your room (i.e. your bed) and draw those sizes on paper too. Cut them out.
Move those cut outs around on top of the layout of your room and see how you can make the most space available.
Then decide where the cage will best fit and what shape cage to make. You can put the cage on top of something else or use grids to make a base and store clothes, books, or other items underneath the cage.