For guys, the end of a relationship is often hard to admit it happened, especially if it's the woman who broke off the relationship. Conversly, same could be said for the woman. If you really want to send a letter, do a rough draft and come back to it the following day and read it and see how it feels as you read it. Does it sound clingy, begging, stupid, in any way?
Then do a re-write and make sure you stick to basic good things about your past together, and let her know that you will be there for her if she needs you.
Then go on with your life, and most likely, you will meet someone else later on. Take the 'turn the page' test. Go buy a copy of Playboy's Girls of Summer or whatever, but one that has page after page of women.
You'll look at one and think, wow, she's gorgeous, then turn a few pages and see another one who is even more attractive to you. Could you honestly settle for just one of them? Turn the page.