It's her own thoughts about you being her 'best friend' and then dropping her that will hurt. You can't really control her thoughts. And you're right, very few people can be rescued, though she's attracted to you because you have offered some rescuing behavior by volunteering to help her out in the past. So there is probably no way to avoid hurting her feelings. The best you can probably do is to be polite and grown up.
If you really want her to fall out of your life (and look closely at your motives to see if this is true), I'd start by not returning her calls. She will probably connect with you at some point, so you'll need to be clear about your needs, which are also legitimate, even in the face of a needy person. She has a lot to deal with, and she may have trouble making or keeping friends. So even a little kindness probably means a lot to her.
The next time she asks for something that you don't feel inclined to give, try this process that really helps us kind-hearted, can't-say-no types:
Use a calm, friendly voice. Restate her request and acknowledge how she would benefit from her request, so she knows you heard her. Use the magic word "AND" instead of "but," which will put your reply on equal footing with hers without invalidating her. Then state what YOU need. This can be as brief as. "And that doesn't work for me." Don't make any excuses, no matter how badly you want to. Don't apologize any further than a brief, polite, "I'm sorry." To do either weakens your calm, authoritative position.
So you might say something like this: "Jeannie, I hear how much you would like me to _________ for you. This sounds like something you would find helpful. And no, I'm sorry, that simply does not work for me."
If she wheedles or argues or cries, repeat the three simple steps. Do not get drawn into an argument, or into making excuses. Keep your voice gentle and calm. If she escalates, which is rare, excuse yourself with, "I have to go now," and either leave or hang up.