Well, you started out with a bad way of getting to know your child. Everyone deserves privacy and if you respected her you would not be snooping. Keep in mind that everything a parent does effects the child psychologically forever and if you are overbearing it will always effect her. People with overbearing mothers tend to feel desperate for attention, the result being overbearing behavior of their own. They tend to be unable to do things on their own, and look fruitlessly for someone who will love them the same as their mother does.
You need to let your daughter come to you instead of imposing yourself in.
I don't mean to be so harsh. I know that it's painful. It's painful for the thing you love most to be suffering and growing away from you, but if you won't let her grow away, she won't ever be an adult and your advice won't do any good if she isn't ready to hear it. I don't know if it helps at all, but I assure you your daughter would be a drag if she never suffered. Anyway, no matter what you or she or anyone did, she'd still suffer. Nobody never suffers. It's the human condition. Amelia and Alex suffer too. It isn't a bad thing. We need to suffer in order to grow and in order to feel joy. Usually a kid goes through some kind of depressed period during their youth. A lot of people get it in high school. I was at my most depressed in sixth grade just like she is. I cried all the time and I didn't talk to people at all. I didn't even want to. I just sat by myself all the time. Now I'm older have friends and a job and sometimes I'm suddenly so happy it's like I'm drunk even when I'm not, just by the way the sun makes it all so golden at dusk. That's what Baudelaire was talking about when he said to be drunk ceaselessly. I still never feel like a winner. I still always feel like a loser and a freak and the people I know are losers and freaks and we talk about interesting things and the people who were and still are winners are talking about the same shallow boring trash they've always talked about and always will, sitting at their televisions, suckling at their devices, blindly consuming. All she has to do is turn her back, care less, and go find something better. And she will. She'll find something so good in herself if she just keeps looking into herself that is so infinitely better than long legs that she won't ever think of Amelia's long legs again except to see that Amelia has something to make her lovely, just like everybody does, and then she can learn to appreciate the beauty of others instead of competing with them like she's always been taught to do by the media.
EDIT: Just to give my input on the idea of medicating your daughter, just please think hard before you do. There isn't one age where mental illness begins to show itself. Different mental illnesses tend to appear at different times and it also varies based on the person. If you have the money to bring her to a therapist, I think that's a great idea. Just keep in mind that if you bring her to someone and they're trying to prescribe her something, depression meds have the extremely common side effect of suicidal thoughts and behaviors in young people who take them. You'll read that on any bottle and hear it stated on any of the commercials. Still, doctors prescribe them. Doctors prescribe a lot of addictive drugs to their patients. Obviously not all doctors are corrupt, but this happens a lot. Now, I've known a lot of kids who got medicated for things like depression, ADD, and insomnia at a young age and they all ended up more depressed, anxious, and restless than they were to begin with. When someone is at an age where their hormones are going through a huge change, their moods will be affected. It's uncomfortable, but medicating away a natural change in a persons development is harmful to them. I knew a girl who was taking a couple different medications in middle school and still looked like a 3rd grader. She was very emotionally unstable and had a lot of trouble in school. In high school she was taken off of her medications and by senior year she was a fully grown, coherent woman who was able to graduate with her class.