Just a few ideas:
Try neutrals (white, grey, cream) that won't clash with the walls (more colors in such a loud room = bad) and put as much of that in as possible.
The bathroom shouldn't be a problem, many people have wild colored bathrooms, all you have to do is keep the door closed.
Personally, I'd pick the colors that went best together and that I disliked the least (i.e. the blue and purple for me) and then try to do something sneaky like cover the remaining walls in a soft neutral fabric (can be pretty gross if you do it wrong, but can also look really snazzy).
With a room that busy, your best bet is just to keep things as simple as possible and draw attention only to the features you do like.
Is there any chance you could convince the family to let you paint the room if you promised to paint it back when you moved out? If it's for their kids, the children's tastes may have changed after two years and you could save them some work in the long run and keep yourself from having to live in a kaleidascope room.
Enough Changing Rooms-inspired rambling from me, hope some of this helps.
Greekgrrl