Sounds like your DD needs more support than she is getting. I can understand your concern about Prozac as there is a risk with all ADs that they will make things worse for a time, especially with young people. There is also the risk of increase in suicidal thoughts with young people, BUT it does sound like she needs medication for her depression and anxiety. Did the GP give her a follow up appointment?
I totally understand that she doesn't want to get out of bed (I suffer from intermittent depression which is severe at times) and the last thing I want to do is get out of bed. My son also had depression when he was 18 and shut himself away in his bedroom too. It's a need to withdraw somehow, as we can't cope with normal interaction - it's what animals do when they are in pain. BUT the isolation doesn't really help us - I used to have a deal with my son that he would get up by 2.00 p.m. and generally he kept to this and also an agreement that he wouldn't lock his bedroom door. He took meds and got better within about 6 months.
Loss of some sort is usually the root cause of depression and anxiety seems always to go along with depression, and it's sometimes difficult to tell which is which. You say she's on her phone 24/7 (like most teenagers!) so at least she is making contact with someone - do any of her friends visit, or call her, do you know. I know it's easy to say but I think you maybe need to try to contain your own anxiety as this will make her worse I think. She clearly doesn't want to talk to you about her problems and you have to accept that, BUT I think maybe it's time for a bit of "tough love" - so you make a deal with her that she gets out of bed by early afternoon - is she eating? Presumably you are taking food to her bedroom - if so, then maybe try and get an agreement that she has breakfast and lunch in her room but gets up for evening meal. She really does need to be helped out of her bedroom, but you have to understand that it can seem like climbing Everest - honestly. Depression is a real torment and can only be understood by those who have experienced it.
Is she interacting with anyone in the family, her dad/brothers/sisters. You say you are worried about giving her the Prozac but what does she want - that's important for her to be involved in the decision. I think she does need medication but you'd need to keep a careful watch to make sure she was not getting any worse - they take 2/3 weeks to "kick in" and sometimes you get the side effects before the benefits. ADs are all a bit hit and miss really, - trial and error.
Have a look at Young Minds website - DD might find it helpful too.
You ask how long this lasts - there's no answer to that - MH issues aren't like getting flu - it's more complicated than that and everyone is different, and there are fluctuations, 2 steps forward and 1 step back - can be a long journey. Maybe you need to find out more for yourself - the MIND website is a good place for information and there are loads of books on depression and anxiety that you could buy to help you understand better this really distressing illness that your DD is suffering............but the chance are she will recover and life will be good again.