If we are to accept that elderly people should have the full say about their care. A view I absolutely support. Then they have to make sensible decisions about how they will be cared for.
I think we need to be more prepared to spell the situation to them in words of one syllable and let them live, at least awhile with the results of their decisions.
In a case like this the lady has to be told that while a family member may able to pay a social visit now and again, there is no family member able to care for her, or render any other service like shopping or cooking. She should be told that unless she is prepared to pay, she will get no other help and she will have to do everything; shopping, cooking, cleaning, personal care herself. If she still refuses to pay. Let her experience it for a week or two to see how she manages.
As we go through life we have all been in situations where the decision we have to make, is not the one we want, but the one that is in our best interest, which is not at all the same. That realisation and decision making applies as much to older people like this lady as to younger people.
However, the reason some older people seem tight, is because they are terrified of finding themselves penniless and will not touch any savings because they are there to be used if a rainy day comes, except when you suggest the rainy day has arrived, they always say that this is only a light shower, what happens if a really rainy day comes. If someone is able to talk through the financial side it might help.
But over all, if this lady is in her right mind and after having all the consequences of her decisions spelt out to her , still refuses to pay for the care she needs. let her manage without care.