I'm in a similar situation to your mother's. Not eighty but getting up there, not on my last legs, but on my "last cat", my last pet. Poly has been my companion since December 1996. Don't know his exact date of birth, found him in the street, and he appeared to be just out of kittenhood. So he's a very sweet sixteen plus. If you credit five cat years for each one, we are about the same age. I have always had animals in my life, but like your mother, I've decided not to get another pet when Poly dies.
I don't want to be tied down with the responsibility for another living creature. I'd like to be able to do some traveling without worrying about pet care. Let your mother's decision stand. Her old cat is still "ticking", and when it dies, she will surely be sad, but I suspect also relieved. What happens after that will be up to her.