Question: Help! My recuperating outdoor "stray" won't use the litter box
Answer: Any cat that is in pain will often have accidents outside the litter box. With the serious surgery your cat has had, she may have difficulty getting in and out of the box. Does she know what it's for? Has she ever used one before on her occasional visits inside? Cats normally prefer cat litter to other textures, and will use it, however they usually do not like to have it right next to where they eat. You might want to take the litter box out of the big box when you are home and gently put her in it. See if she has a problem getting in and out of it. It might be too high for her, so either try a lower box or cut an opening low enough for her to use.
Also, when you clean up cat urine from the carpet, don't use an ammonia-based product. The idea is to remove the urine scent (which ammonia mimics) -- vinegar and water or plain water will work, followed by an enzyme-based product that will neutralize the scent so she won't return to that spot.
Finally, since you apparently love your outdoor kitty enough to have paid for surgery, please consider adopting her "officially," and training her to be an indoor-only cat. I'm guessing that she met with a vehicle accident. You surely don't want anything like that to happen again.
For more information, please see this related FAQ.
More Cats Q&A