Re: Awful carpet.
If it's as bad as you say it is, there is nothing you can do topically that will remove the odor. Once urine has reached the carpet backing and pad there is no cleaning it.
As dj1 suggests, wear protective clothing and a good dust mask and pull the carpet. From there you will want to clean the floors thoroughly, then apply a good coat or two of pigmented shellac to seal in the odor. Cats also have a very nasty habit of spraying the walls, so you will want to carry the shellac up at least two feet, if not all the way to the ceiling. The main reason to go to the ceiling is that the shellac will give whatever top coat you use a different sheen from the rest of the wall.
Once the floors are sealed you can install whatever type of flooring you like. If you still have those nasty cats, I would recommend a solid surface such as tile with well sealed grout.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!