If the letter admitted things were final, and tied up all the loose ends and things, I think it was quite sweet. Even if I didn't appreciate it at the time if I was still a bit bitter, I wouldn't throw it away and when it had all blown over, I'd enjoy having a final marker of the relationship. Especially if we'd decided to stay friends, I'd like it then.
If on the other hand, it was trying to make me feel guilty or win me back, I might be a little irritated. If they want to sort it out, they could talk to me face to face, if not, don't try and guilt trip me!
My personal take on things.