If I gotta go out I'm going out fighting. It's not like I've been just sitting around sad these 10 days. I've been hitting the gym, getting money off my brand, creating, hitting some hoes just for the sport... So it's all good. I'm honestly prepared for the worst.
You are setting yourself up to fail with this girl. Ten days is nothing. Particularly in the summer - days just fly by with things to do. She missed you after 3 days, then by day 7 she is just annoyed (or relieved) by your absence.
At day 10 you are probably going to catch her in the middle of her angry phase, when she'll be the hardest on you.
Conversely, if she legit wanted you back and has been holding out for you to change some behavior, she knows you have not changed in 10 days. She knows you are just impatient.
You won't know which stage she's at because your impatient ass wants to send a letter. Which puts the burden on her to decide how to respond. And if she's of the latter camp, where she wants you to change and knows you haven't in 10 days, she will be waiting on your ass some more. But you will have given up.
I don't know what you are thinking, you know this girl better than we do, but this does not sound good for you.