Hi, Susan. You have a common problem and one without a really great solution. You are correct that sanding alone will generally not cure the problem. The best I can offer is to start by stripping or sanding the remaining finish off. On the black spots that remain, try wood bleach (oxalic acid). Follow the label directions and particularly the safety instructions. Baking soda will neutralize the acid if you need it, so have some around just in case.
If it is as bad as you say, carpeting or an engineered floor may be your only solution. But it will not hurt to give it a try. Since you will want to seal the floor even if you put carpet or new flooring down, you will have to do the whole floor refinishing routine. This will keep any smell from coming back later as well as protect the wood from water damage.
It could be worse. A friend of mine rented a place to people who cut a hole in the living room floor so they could throw their beer cans in the basement. They took 5 pickup truck loads of cans out and the smell was equal to or worse than your kitty purrfume. Good luck with the oxalic acid and let us know how it works. Henry in MI