Nature's Promise makes a spray that gets rid of cat urine smells, it breaks down the enzymes in cat urine. You can also get products that absorb odors, such as activated charcoal bags, idk if they work for cat urine though. It doesn't seem like there's an easy answer for you. I don't like the dog in a cage all the time, that's horrible, but idk the circumstances. I did thought that rescues or fosters had to have cats spayed/neutered, that would stop the marking, but again, idk all the details. I would Google products you can buy to help eliminate the smell to your home. Cat urine is strong and will definitely aggravate asthma. Good luck!