ah. I see. I thought you meant she'd walked away from you because you had children.
All her 'you didn't bother enough suggests that you did make contact, she just wanted more attention than you were able to give at that time. Yes, you could have kept in more contact, but tbh, I would expect a grown up to be able to understand that lives get busy the older you get and you don't need to be in each other's pockets in order to be friends.
You have apologised more than once, you have accepted your part in it, hell, it reads like you've practically got down on your knees and begged for forgiveness. If she wants to sulk like a five year old then let her. I am not sure what more she wants from you. For you to strip naked in the town square and submit to a public flogging?
At this point, I think you have to accept that you have done all you can, you have tried to set things straight and she is more interested in a load of 'oh poor me' crap than actually allowing you to mend fences.
And you're right - the you talking to someone who used to do whatever is really pathetic. Unless of course, that person spread such vile things that it really ruined her life, in which case I can see why she'd carry hurt with her.
She doesn't want you in her life. You have to accept that and move on without her. Just be civil if your paths cross and stop trying to plead for forgiveness.
Just out of interest, she wrote to you saying how you hadn't made any effort - how much effort did she make? Did she contact you regularly? Or was she sitting back huffing that you didn't contact her enough?