I'm no expert of how to approach the weight problem with her (my mum just kept nagging me I was fat, even thought I really wasn't), but how about changing eating habits together with her?
However you look at it, she needs to eat much less sugar/junk food/fizzy drinks, etc. Not talking about cutting food groups or treats completely, but eating less of it. Start eating much more veggies, healthy fats, fish, pulses, eggs, beans, some fruit. Skipping mayo/ketchup and the like. Less bread, pasta, chips, less potatoes. Also, don't be tempted with 0% fat food alternatives, fat is not your enemy, sugar is. Food companies eliminate fat from such products, but replace it with heaps of sugar for taste. So it's less filling and way more harmful than a normal full fat alternative. Plus seemingly 'healthy' foods: flavoured yoghurts, granola for breakfast, juices and similar - full of sugar.
Maybe you could tell her that YOU would like to lose weight and improve your health by making better choices, and maybe she would like to help you/support you in this? Just a thought. As she's a young girl and won't have enough will power and would feel hurt if she'd alone would be served 'rabbit' food and rest of the family eat 'normal' stuff.
About rabbit food. It looks like that from the start. It's difficult to change and it's not tasty at first (at least it wasn't for me). But if you persevere, you'll get used to it and will like it. I love my 'rabbit food' now, I really do :). Benefits to your health (and looks! :) ) are immense.
Also, get her to a GP, of course. Like other ladies said, it could be PCOS or some other issue.