Using a Rug Doctor cleaner on Cat Litter?
I don't think the carpet cleaner will make it worse, but the urine may have soaked into the padding subfloor. Could you get a covered litter box. Our cat also had the same prob I replaced the tray with a covered box there is no longer the 'missing the box' prob the litter around the box is only from tracking, not kicking. If you think kitty cannot or will not use a new box, then you could invest in a high-quality doormat type little rug--like the ones people put in front of the exterior doors for wiping their feet. These rugs have water-proof backing so the urine will not soak through. Then all you have to do is rinse it off with the hose outside or wash on 'gentle' in the washing machine line dry. The backing will eventually crumble by washing it in the machine, that's why I suggest hosing it off outside. But you could also use Rug Doc to clean it periodically or invest (or ask for X-mas or your b-day) for a carpet cleaner or Spot Bot. Anything else will just get soaked through still get to the carpet. You could always put a tarp under the litter box. Sorry about the long post. I have 8 cats have been dealing with cat problems for the last 7 years. I found what works best for me I like to share the info.
Move tray, try a dry carpet cleaning powder. Foam cleaners work well if used correctly. Deodorize and vacuum regularly. I put old newspaper under cat tray in case the cat as an accident.
I think shampooing will remove the stains.