If the cat's back is broken, euthanizing it is the best thing. I'm an absolute cat lover, but if one of mine had a broken back, I'd put that one to sleep without a second thought. It's not like a human who can still live a fulfilling life. The cat won't understand why it can't move around, will be incontinent, etc. Of course, I'd cry my eyes out afterwards, but I'd do so knowing I'd done the right thing.