Contact her through social. I think that's a better option than writting a letter. But the letter could be nice too depending on how close the two of you were and how the no contact unraveled. If it was nasty and hostile, I would refrain from sending a letter. If it was clean and soft then maybe I would send it. Look at all the surrounding circumstances. Take note to every angle, weigh your odds and figure out what you think would be best. But honestly I think social is the best shot ya have.
Well, if you are sure she hates you now, then there is little chance you have of changing her mind. However you can try if you want too. I think you should go and talk to her in person. That will be a better idea.
No it is not a good idea to write her a letter, I understand it is hard, but you should at least put her aside and start seeing someone else. See how she reacts. If she really has deep seated feelings for you, then when you bump into each other again, try to gauge her feelings for you. If she acts hateful towards you then you know that you are seeing someone else who really cares for you and it doesn't matter, she isn't in your life anyways. If she acts sorry and sympathetic then you know maybe now is a good time to write her that letter (if that's your love language). The most important thing is not to put her on a pedestal. Nobody deserves that and you will just be run over and she will stop respecting you which just turns into an even bigger problem. I speak from experience.
Why I say you should see someone else is because from my experience, this will put things into perspective for you and it will be a way of not putting her on that pedestal.
If you want to write her a letter, write it, but don't give it to her. Just write it so you can express your thoughts to the one person that will always have your back, if you let him, yourself. If you give it to her, she will probably act hateful towards you and put you on the verge of suicide.
Good luck. I have been there so I know it's hard. 😞😔