Anorexia is not about weight, it is purely mental with physical side effects. If you are feeling the need to purge and restrict, feel you do not deserve/should not eat, and are getting scared by food, then you need to go to a doctor because these aren't healthy thoughts.
The doctor can refer you to a clinic or some sort, but there are also (quite expensive) private options. The NHS should not say your BMI is too high to receive treatment because a) it's not, at this BMI I was still at the beginning stages of my refeeding programme b) they're not idiots and realise that with eating disorder patients there are a million other health problems aside from weight that are as a result of the disorder c) they prioritise by health (or so I got told. I think it's like if you're suicidal, have had heart problems, if you've lost your period) NOT by weight. My Mum rang up the clinic many times a day until they agreed to see me. Push it.
You will need to be completely honest with your family. It's
very hard to distinguish between anorexic motives and your 'true' motives. Eating disorders will justify every single self-destructive move they have in store for you. If you feel a true sense of desperation to conceal something from your parents, then straight up tell them what you're hiding from them, because that sense of desperation is coming from an eating disorder.
Don't let yourself do it alone. If I had any advice that would be it. Surround yourself with people who love you, be completely honest, and every day they will begin to understand more. At first my Mum thought she understood but in reality knew NOTHING, but now she knows when I'm struggling just by my body language and we sit and talk it through, and afterwards I always feel a little better. The best way to conquer this is with a huge support system under your wing, people to tell you that certain behaviours are irrational, people to tell you that actually, eating a 1000 calorie pizza is bloody marvellous and is absolutely nothing to be ashamed of.
The thing I did which no one else was involved in was created a tumblr recovery account. These can be SO triggering if you follow the wrong people, but I really believed my tumblr account saved my life.
There are AMAZING people on there, they send you lovely messages of support and love when you need it, and the best thing I found was that food actually became normalised! I talked to people from all over the world about the foods we eat and how much we love them, some of them eat up to 4000 calories a day to recover (quite a few ate a pint of Ben and Jerry's every day to make up the numbers) and they share pictures, it's really comforting. I rekindled a love of food from tumblr, it helped me so so much!
In terms of what you should be eating etc, a professional should really be seen here. For me, I didn't count calories or whatever, and I felt I was eating loads, but then my Mum counted one day and it wasn't as much as we thought. For my recovery I had AT LEAST 2500, usually nearer to 3000. Don't worry about what you'll do once you reach a better weight, it'll naturally take care of itself and once your body has been refed (ie is healthy) then everything will start to normalise.
You're gonna have to be brave, and it's not going to be easy, but there is so much life in you yet and it's not worth spending it on an eating disorder. 500 calories a day is slowly killing you. You are worth recovery and I promise you it is the BEST choice. I've actually come through better than before I went in!! I'm so much more confident now, I have better social skills, I'm more optimistic, really enjoy food, eat a lot, value life more, and yeah I still struggle at times but
I am in control of my life now, not my eating disorder, and that's amazing feeling. Good luck, you can do this