Admissions are done on a holistic basis, which makes it very difficult for us strangers on the Internet to be able to assess your profile accurately, since a lot of it will depend on things we don't see here, and there aren't many reliable quantitative metrics that we can use for what we do know!
Quick note: You should seek advice of your LOR writers to determine what to divulge about your illness. Catria gave a good advice to have them write about it instead of you. Another thing I can add is that you can also choose what parts of your illness to focus on and what details to provide. I think it might be too personal to say it was an eating disorder, and even though it shouldn't be this way, the reality is that mental illnesses have a lot more stigma than physical illnesses. So, one way to frame your illness is to discuss the physical aspects only (hospitalization, but don't say what for, and the broken hip).
One way to approach this is minimums. Although you may not be interested in a top 10 and your chances may not be great, you do meet the minimums for entrance to such a program. They usually have a 3.0 GPA cutoff, but the competitive GPA is usually much higher. However, the rest of your profile is really strong:
1. You are mentoring a student (this signals that someone trusts your research and regards your ability highly, which I think will mean a very strong LOR)
2. You are going to have two first-author papers (many grad students don't get to this until their 3rd year or so of grad school)
3. You have a lot of actual research experience (being at a top school now, it might seem like a lot of other people have experience too, but the reality is that most grad school applicants have very little experience)
4. You might be able to get your LOR writer to explain the impact of your illness on your GPA, and
5. You are getting a degree from a top private university.
These five other factors, plus a high score on the PGRE, might be enough to mitigate the lower GPA (assuming you can also raise it a bit this Fall). I don't want to instill false hope, so I will say that even though there are good reasons and even though it sounds like you are an excellent researcher, the low GPA could make it very difficult to get into a top 10 program. And, I'm assuming the most optimistic possible case for the five factors above.
But, if this is what you want, then I don't think it's "out of the question". You will have to cast a very wide net when applying to programs and do a lot of legwork to determine best fit programs. However, there's no point going through all of the money and effort to apply to top programs if that's not what you want. I'm just saying that if this is your dream/goal, then don't limit yourself because your GPA is low--you might not be successful in the end, but it's worth the effort.
You definitely have the right experience and abilities to succeed in a graduate program though, top 10 or otherwise. Find the programs that interest you, consult with your advisors, and apply to them!