Your poor daughter. It sounds like an awful situation for both of you. I second other posters saying that counselling and possibly medication would help. Other than that, as this has been going on for so long she likely has incredibly low self-esteem, which is where the 'don't look at me' and such probably stems from.
I wouldn't advise 'tough love.' She doesn't sound strong enough for that, and she needs to feel as if you and she are on the same side. In my opinion, it's important to recognise that for her, anxiety does make the world a terrifying place. It's completely irrational, yes, and in the long term you want to change this, but it's important to start from where she is and not where you'd like her to be or where a 'normal' young woman her age would be.
School isn't for everyone. You could try talking to her about what she'd like to achieve, in a way that lets her take the lead (so no 'you should' or 'you must' statements if you can help it.) If she hasn't got GCSEs, could she work towards Maths and English using an online or distance learning course - perhaps in conjunction with something more vocational if she's interested in it? If she has any craft skills or is business-minded, could she do a little bit of Ebay selling? Not as a full-time job, but as something to boost her confidence that she can do without leaving the house. If she's having difficulty leaving the house at all, enrolling on a full-time college course is only setting her up for more failure.
Obviously, it'd be better if she could go out! But this really needs baby steps - to the end of the front garden, to the end of the street/local shop, and building up gently. If her appearance or people seeing her is causing anxiety, she could wear a big coat, or leave her hair loose and hanging in her face a little - another temporary solution, but confidence really can't be built overnight. If she likes make-up, that's another way of presenting a persona to the world that isn't necessarily 'her.' I wouldn't comment on her weight to her, though - yes, it's obvious that if she was more active she'd be slimmer, but the anxious/depressed brain will interpret this as 'you're saying it's my own fault I'm fat, because I'm too anxious to go out and exercise - another reason to feel bad about myself' rather than seeing it as a well-meant suggestion or neutral comment.
What has she been doing at home? Whatever it is, you could talk about the transferable skills and good qualities she's developed from it - and there are some, even if she's been spending all her time reading or playing video games. Someone who feels anxious and as if she can't even go out doesn't need to be told again where she's gone wrong in life - she already knows.
Sorry for the lengthy post, and obviously not all my suggestions may be relevant to your circumstances, but I was highly anxious and out of school for long periods myself at that age, so I know what worked for me and what didn't. Seeing a psychologist was really, really beneficial for me, but only in adulthood - before that I was treated like a child and the therapy was focused on trying to get me back into school rather than addressing any of the deeper psychological issues causing my anxiety. As a result, I didn't trust my therapist very much. The other thing that helped me was volunteering - I got to prove to myself that I could be competent and help other people, and the organisation appreciated that I was giving my time. For me, it was a necessary step on the way to paid work. (I chose to do this, though - someone telling me that I had to would have led to panic and withdrawal).
I eventually held down a full-time job for more than 5 years, then left it to go to university - better late than never! So don't lose hope for your daughter. These things can get better, even if they're incredibly frustrating to live with at the time.