It could be a UTI, as opus suggests. If a kitty is not feeling serene, he or she might not keep at it with regular water drinking and so on, and could get sick. I would consider getting rid of the rug entirely as I will bet the smell will always be there, no matter how many enzymes you use. And if kitty is not well, put down a piddle pad there for a while. (Large sized human-being incontinence pads are great, and for some reason not as costly as the versions sold at the pet store.) But do take kitty in to the vet. I had one cat who always announced his UTIs by peeing on the windowsill.
Miss Teia poops in the litter box, but only pees there part of the time.
She does use the puppy training pads I put down and has peed in an inappropriate spot only once since stating use of the puppy pads over a year ago.
it could be a combination of both....she is upset with a new animal in the house and his scents all over...and is telling you!!
also with an 18 yr old kitty, it can quite likely be medical....to be safe I would take her in for a urinalysis and make sure they check the PH of her urine too. it could be crystals or a UTI.
when it hurts to pee, they tend to let us know that by going in inappropriate places...
the pain can also be with her bowels, is she regular and not constipated? pain ANYWHERE in the bowels or bladder will cause a kitty to seek out another area to relieve themselves b/c they associate this pain with their litter box.
also be sure not to just wash the rug, you need to clean it with an enzyme cleaner(avail at petstores)....the enzyme will help remove the scent of the urine, remember even if you can't smell it they still can with their extreme sense of smell..and if they can smell urine they will automatically think its an 'approved' spot to keep on using
good luck