Hello,
It is admirable that you admit that you want help, and you are willing to reach great lengths in order to ask forgiveness, and make a relationship work.
It is hard, but not impossible to rebuild something that is broken. You need to show that you understand your mistakes, regretted them to the greatest degree possible, you have really changed, and will never make the same mistake again. You are still the man she fell in love with. The apology should be deep, romantic, with the words coming straight from your heart, touching her soul.
In order for such an apology letter to be written, more details are needed. As many details as possible, in order for the best setting to be set.
Some of those questions could be:
-What are the mistakes that you are apologizing for.
-Does the girl like romantic gestures? Should the letter be romantic with impeccable and exaggerated vocabulary, like a poem, "your melody rejuvenates my body, your silence sunders my soul", or it should be in a more realistic tone, with no poetic metaphors "you are everything I want and everything I need, every moment apart from you makes me realize how much I have lost, because you are the world to me"?
-The best moments that you have spent together, and what made your lasting relationship so beautiful. Remind her what you were together, and what you can be when you are together again.
You can contact me on freelancer in order to discuss. We can write a letter that will touch her soul, make her reconsider.