I'm not going through anything like this but I'll offer my thoughts anyway if you don't mind. I don't think that your daughter needs to accept your boyfriend right now. You say she is having her own problems and that is enough for her to deal with. Have you suggested meeting her at a neutral location for coffee or something? It would be better than trying to talk to her over the phone. Also, I would not bring up your boyfriend to her in any respect. He is your business. And no, I don't think (just my personal opinion here) that you should go back to your ex to appease your daughter. It just would not work out. My parents separated when I was in my 20's and though it was hard, it was a relief not to have the stress of them living together in the same house. They should have separated long ago.

Try meeting your daughter for a shopping trip and lunch or something. And keep the focus off of you. Maybe just keep the conversation light such as discussing her schoolwork, friends, news headlines, etc. Let her raise matters of personal troubles at her own time. Don't push it. She will notice and appreciate the respect you give to her. Have patience and don't give up.