It's a sweet thing to do in that case - but I'm not sure I'd go with the letter -- and I don't mean this to sound as mean as I think it's going to -- I just can't find the right words - it almost comes across as needy.
She broke up with you. She hurt you. You are undestandably upset. Why should you try to make her feel better? (I'm really trying to work on being nicer and showing more compassion towards others.... it's hard
If you think you can be friends with her, that's cool, just remember your feelings too... Figure out exactly how you can be friends with her - then go from there...