An apology says that you regret what you did and that you'll try very hard for it not to happen again. Part of why it is an apology is that it is uncomfortable, so that you are not good at this just means it is time to get some practice in. LOL!
A spoken, face to face, apology is better than a written apology, because you and she can both read the body language. She can see that it is uncomfortable for you but you are doing it anyway, she can see that you are sad. You can find out immediately if you are forgiven and not wonder will she even read the letter or when will I hear from her.
So tell her in person, promise both her and yourself that you'll try to do better.
I always think that if I had never done wrong then maybe I'd have the right to bear a grudge, but I've done plenty of things wrong and hurt others, and to get their forgiveness makes me feel better. So I forgive also.
at 6:01 PM on Sep. 8, 2009