It might help to write an email, setting out that the relationship is definitely over but you would like the girls to have the best possible relationship they can with their dad. And that all depends on him.
Don't send the email, just write it all down. Then sleep on it and come back to it when you're ready. Make any changes or adjustments. Get it sorted in your head.
If you suddenly find yourself in a face to face situation with him, you will be prepared, having already thought all this through. If not, keep coming back to the email and check that you are happy with it before you send it.
Take all the time you need. If he's not that bothered about contact right now, take the time to just mull things over. Mediation would probably be helpful provided that it is very clear that you only discuss contact with the girls and, if necessary, financial matters. Make sure that the mediator is aware of this and is prepared to steer the meeting accordingly.
Can't remember if I've mentioned this, but let the school know. Are you happy for him to collect them from school? Is that something he might have done in the past? That's something else you will need to agree so that he doesn't just turn up one day and take them. Of course, it's fine for him to do that, but it should be pre-arranged so that everyone knows what the plans are.
Whilst he may be seeing ow now and again, I bet he hasn't told her how he sent you that song link, or how he begged for another chance, or how he would come back like a shot if you'd have him. She is getting her karma but she is not responsible for his actions. She knew he was married and a woman who would sleep with a married man has to have a pretty low opinion of herself in the first place. You should probably feel sorry for her really, but I won't hold my breath 
x