My advice --- Personally, I am too nice to just kick somebody when they're down already, she seems to be having multiple issues in her life currently. That's not to say you should comply with her requests, because IMO, you shouldn't. Don't be rude though, although I am sure you wouldn't be. This is a very sticky situation, though.

I would tell her, no, but of course in a nice way. Don't be mean, but don't let her walk all over you. I think she's using you, in a bad way. If she had proved herself a worthy neighbor from the beginning, I could MAYBE see it, but she was rude and that's not fair. Why come to you now, just because she needs something? I would accept the apology and move on. I know it's a touchy thing you're dealing with here, the whole family issue. But she can get an air card of some sort too. She's not totally out of options.

I wish you luck with this situation.