Dear {Recipient},
I would like to apologize to you for the accident that occurred on {date}. It was entirely my fault, and though you have been more than gracious, I feel the need to extend a formal apology to you. It was absolutely an accident, but rest assured that I will be more careful in the future.
Even though my insurance will cover all the damage to your car, I know this incident has caused you stress and inconvenience. I would like to extend to you a small gift as a token of my appreciation for your honorable behavior. Please find enclosed a {gift} as my thanks.
Again, I am very sorry for having {specific accident}.
Sincerely,
{Sender}
