How do I apologize to her without coming across the wrong way

This is this girl that I have been through a lot with, we've been on and off for almost 8 years or so. I had an epiphany last night and want to right some of my wrongs. We've had a handful of bad times that I feel we're my fault and I never sincerely apologized to her.
How do I apologize to her without coming across like I want a second chance. I'm not trying to get back together with her or anything, just an apology. And I don't have her number or anything so I will have to do this over the internet.

Have you heard of email?

I didn't how like that, I already have a way to contact her over the internet, I meant how more along the lines of what to say.

Sorry for being a dickhead?

I seriously doubt she's been giving it any thought since you've broken up. If you can't think of what to say, then don't say anything. Forgive yourself and move on is probably what you should do.

I doubt she would even appreciate the gesture.

off and on for 8 years and you don't have her phone # that is a bad sign. say " sorry that i haven't always been as thoughtful to your feelings and situations as I could have been, I hope you take me as sincere and i never wanted to make you feel bad" along those lines, how can we supply you with words when we don't know your wrong doings against her the last 8 years?