Similar situation (although 22 not 15): I don't know any solution, but I would NOT suggest birth control pills. The reason is that birth control pills don't actually give you a period in the same way that your normal cycle would: when they were being designed, the makers thought that women would feel more "natural" if they bled every month, so they made the pills so that a withdrawal symptom was bleeding from the uterus (obviously these things were designed by men). So the "period" you get on the pill is not actually a normal period; it's just a withdrawal symptom when you go on the placebos for a week. This is why it's possible to just keep taking the non-placebo pills and not have a period at all for 3 or 4 months. I'm not an expert, but it doesn't seem to me like that would help you get your regular period or healthy hormone balance back.

sometimes you have to gain more than you originally weighed when you had regular menses (one study showed 2.4kg more). also, sometimes exercising/running that much makes menses disappear despite caloric intake. you've got the female athlete triad going on with secondary/hypothalamic amenorrhea. unfortunately hormone replacement has not demonstrated a protective effect on bone density measurements according to the literature. it may help if you have other symptoms of low hormones, but it isn't a solution by any means.

keep gaining weight slowly and cut back on the exercise if possible.

I have searched far and wide for a solution to this problem that didn't involve gaining weight. so far the only other thing that can cure secondary amenorrhea is leptin injections, since leptin is the hormone secreted by fat that tells the brain it's safe to reproduce. unfortunately, this treatment is experimental and not available yet.

You must see a doctor. This is far too complex for random Internet strangers to advise you.