Actually, my personal statement was about my own experience with an eating disorder. But, I was very careful not to frame it as a "sob story," as you put it. Rather, I explained how the experience not only inspired me to take an interest in medicine, but also caused me to reflect upon the sort of doctor I wanted to be -- that is, how I wanted to interact with patients and engage with them the way my doctor did with me. In my experience, this is not a weak point -- I've had plenty of interviews, and one of my interviewers told me that my PS was "very unusual and extremely compelling." If you do decide to mention it, make sure you address that you are recovered! I know there is some debate about what types of personal difficulties are okay to mention in a PS, but I for one think that it is okay to mention EDs (a) if you're recovered (and I mean completely recovered) and (b) it has somehow impacted your views and goals in medicine.
Personally, I would advise against the type of PS that you're considering... I am not saying that the issues and challenges you faced were insignificant or trivial in any way, but the PS is the place where you sell how awesome you are. However, there are some people who do address mitigating circumstances in the PS and meet with success, so it can work. I would write it about some experience that has been crucial in your path to pursuing medicine, and only mention the other stuff if it is relevant to that. If schools want to know about the deficiencies in your application, they will ask -- answer as professionally as possible ("I was experiencing significant personal challenges, which I overcame and then went on to get a high GPA in subsequent semesters"). Good luck! (and if you're looking for someone to proofread your PS, PM me -- given my own experiences, I'm very interested to see what you come up with!)