Tell her no, that you don't have room in the basement for her or the dog. Of course, you would like to help, but maybe you could help by asking one of those relatives to take her in.
I think something's very odd there. She won't tell you where she's living? Perhaps she's already homeless. Then why not say so? Perhaps she's on the run from the law?
She'll have money from selling her house, so perhaps she can show a landlord her bank balance and rent a small place of her own. Or maybe she can rent a room from someone else who's out of work and looking for income.