The neighbor cats hanging out near your florida room could be stressing out your cat. I'm not sure the best way to get rid of those cats. Odds are the cats outside have peed all over the outside. Main reason I know this is because, when I'm out w/ my cats they stop and smell all the cat pee around the house, the neighboors house, some bushes, and things in the yard.
You could always go out and throw water on the cats hanging out. When we moved in to our current house. There were 2 cats causing some stress for my cats. One the next door neighbor cat, and then one across the street. We kind of learned to deal w/ the next door neighbor cat. The cats peed in each others yards claiming things. Things didn't calm down until our cat, put their cat in its place. They had a little roll, no injuries occured, but their cat kept away more. Plus, when we wanted him out of our yard, squirt bottle. Of course with him, you didn't have to get him wet, just present it, and he left.
The other cat, when we would see it, we would spray it with water. Then eventually we would just look at the cat, and it would run. I'm a big cat lover, so its hard to do, but you have to protect your kitty and realize it can cause stress.
Definitely check w/ what the vet says. You can give the cat prozac and other calming drugs.
We had a problem in the new house, peeing in the basement. It ended when we added a litter box downstairs. So, temp soln maybe to add a kitty pan there. It may help. A kitty box would be much easier to clean out than a random place the cat is peeing.
We use simple solution to get out the cat pee smell. Works really well. If you put some in the washing machine w. what was peed on, let it soak, and in addition spray the spot too.
Same cat pees in his cage when in the car. So, we spray the blanket or cushion down, and then put some of the simple solm in the washing machine along w/ the regular detergent. It works if you get peed on too for cloths. Namely when the 2 cats decided they weren't sharing the same crate on the way back from the vet, and pulling into the driveway, got peed on. Better me than the car seat, thats way tougher to get the smell out.
Same cat. When he would run away, and disappear, into the woods behind our house (he is suppose to stay close to the house, he now has a run to be in when we aren't directly outside watching him. )
So, he comes back, and so as punishment he got a bath. Well, after the bath he went into the second bedroom and peed on something. He sometimes pees while getting the bath too. So, we figured he was doing the peeing cause he got punished, and he was one upping us. We tossed the been bag, and his old bed he use to sleep on in that room.
Have you punished the cat recently in a new manner that he is acting out against, any other new pets in the house, or new family members?
Good luck, keep trying different options!