Either plan to spend a ton of money on repainting it or just toss it out and get a new one.
Sand the entire thing, then use a super strong primer. I recommend going to Lowes or Home Depot and talking with them about what primers you should use. Something super strong meant for bathrooms to kill mold might work. Might. The ammonia in cat pee is super strong, and once its dry it doesn't go away, so you have to use the primer to essentially hold all of the toxins and smell in.
You will need to fully sand the entire thing, inside and out, then use the primer. I would sand between layers of primer and do a few good coats to be sure it stays inside. Before you paint over it let the primer sit for a while so you can be sure that the smell isn't there anymore. It would suck to have to redo the entire thing because you didn't add enough layers of primer.
Or, like I said just get a new one. Cat pee is almost impossible to get rid of once it dries. On a porous surface, like wood, you can't get rid of it, all you can do is hope that the primer holds it all in.
If you have a cat, throw it out. Even through all of this your cat will still be able to tell that the old cat pee is there, and will start to pee on it to cover up the old cat pee smell and put theirs on top. Its a dominance thing, and not something you can really control or prevent, other than just not bring the cat pee covered furniture into your home.