I like LazyDaisy's wording for your email.
You said you're okay with being a "distant acquaintance," so I guess you have to figure out exactly what that means for you, especially if she's calling and emailing you a lot right now. For example, do you want to exchange emails with her? If so, I don't think you need to reply to every one she sends you, and not right away. When you do reply, keep things somewhat short, just a paragraph or so, and light. Touch on a few things that she said so you're not entirely ignoring the content of her messages; but maybe say things like, "Oh, that's too bad. That must be very frustrating. I hope things turn around for you soon," as opposed to getting sucked into offering suggestions. Add in a few light things about your own life, send the email, and then decide that you won't reply to her again for, say, a month, no matter how many emails she sends you in the meantime.
I don't know... Does that sound polite to people? I guess I'm having a problem with the "keeping her as a distant acquaintance" thing. I don't think I'd want to "keep" her as anything, and I would be afraid that replying to her emails--even just once a month, with a paragraph--might be too much encouragement. Honestly I think I would go with MsMarjorie's suggestion--tell her I was too busy to make any plans for the foreseeable future, never respond to her again, and hope she stopped contacting me soon.
Also, I personally wouldn't address her negativity, or difficulty in making friends, unless she specifically, sincerely, and repeatedly asked me about it. I wouldn't consider her a good enough friend that I would bother trying to help her "get better," you know? But that's just me, and I don't think it would be rude to mention those things, if you wanted. I personally just don't see that it would help anything.
ETA: oceanus has an interesting point. I will give that some more thought.