Break-ups are so painful - I'm sorry you're going through that struggle.
Your question says you broke up with her, rather than the other way around? From what you say, it sounds like you feel you could have handled things better during your relationship?
My suggestion would be to concentrate on treating her better and behaving differently rather than trying to verbally convince her you're different now. If you've truly changed, she will see it in your actions. It may well be that she's not going to return to the relationship, but at least you'll know you've given her the attention and affirmation she's deserved all this time.
I must admit that it hurts women deeply to see the man they love shower attention on other women but not on them. So changing this situation (if it's possible) won't happen quickly. The fact that you have two children together gives you an automatic reason to stay in touch (I assume). Be respectful of her decision as you move forward and stop trying to convince her. Instead, show her.