Many of us here have been at our wit's end trying to help a cat who pees and poops outside a litter box, so we do understand your frustration and misery.

As others have said, a UTI is a major reason for a cat to pee outside the box and since this behavior has lasted for months, she could be in very real distress if it is a UTI. Taking her back to the vet to check on this possibility and to discuss her problems with the vet is part of your agreement with your grandmother to take care of her cats. Illness of many kinds does affect the way a cat will or won't use a litter box. When my Artful Dodger developed renal failure, he stopped using the box and would only pee on the newspaper (over linoleum) around it.

It could be both medical and behavorial, it sounds like it to me, which means that at this point Jill is screaming at you for help. Cats can't talk to us, they can't go out for a drink to relieve their stress, and litter box misbehavior is a major way for them to communicate that they are upset. The fact that she is the dominant female should not be held against her, there are always dominant and submissive cats. What was her behavior like at your grandmother's? Did she dominate the other cats then? Was she destructive while she lived with your grandmother? Jill will sense that you dislike her and respond accordingly, just as we would respond in some way if we had to be around someone who disliked us. Jill can't be "broken" of her bad habits, she's not a dog or a horse, you have to understand why she is doing this and change things

Taking Jill to a shelter would be taking care of her in comparison with "kicking her outside". Since you wonder how you would feel if she didn't come back, it indicates to me that you wish this problem would just disappear, but I hope you'd feel pretty bad if you did open the door and never saw her again. That's not a promise kept.

Take her to the vet. If things don't work out, take her to a no-kill shelter.