Thanks to everyone who replied. There are some good ideas here, and I feel a little better equipped to handle the problem, and a little more certain that it's not insurmountable. At least I don't have carpets, so my floors mop up pretty easily.

I've been using Pine-Sol, which Sophie seems to hate the smell of, so I think that's helping.
I'm still not going to crate, although I don't agree with Eliza that it's inhumane. It's just that for a variety of reasons concerning lifestyle, surroundings, and other pets, I know it's not the solution for Sophie and me.
Anwyay, we'll keep you posted. Thanks again.