Ok, if my maths are correct(?) that puts you at about 50kgs which for 5'1" is fine, especially if you are well toned as alot of the weight (not that there's alot to spare esp.If you were to get sick)would then be in muscle tissue not "fat".
If you like most foods & eat a reasonable amount (which should increase in summer if you are being moer active!!) & none of the foods make you sick (like being allergic to them or them giving you tummy pains) then you should make sure you eat a balanced diet (roughly, I mean you don't need to be gung ho everyday on getting all the food groups so long as overall you get enough of what you need!!) & health/food/eating wise you should be ok.
What as you yourself pointed out you need to work on is self-esteem, depression & self-confidence. The depression may have started after the eating issues but it & eating probably both come from the lack of confidence & self-esteem. Eating disorders don't just happen out of the blue - there is a cause (abuse, bullying, violence, trauma, loss, self issues etc) & if you are to get better fully you need to deal with these.
So please go back to my first post, put smilies up, smile alot, make funny faces in the mirror (this releases chemicals to reduce depression), find a yoga program on tv or buy/borrow a video & start doing it, it will help you keep busy, help you tone, help your health both physically & emotionally. Get a copy of sam obitz' s book if you can afford it ($us13) been there, done that? Do this! On cbt (cognitive behaviour therapy) or see if your local college or hospital run course on cbt they're usually free or very cheap & could well help you overcome your problems before they become alot worse & with your summer coming up you will have some time on your hands when you could do such a course.