I know that you probably might think that this can't be an option but I think that in your situation you should let her go and move on. Your breakup sounds messy and confusing so I think that the feeling of wanting her back that you have can be mostly due to your desire for some real closure to why she broke up with you.
I had this happen to me when my ex broke up with me without warning. I had this desire to go back to him and I wasn't sure why. But months after we separated, I decided to meet up with him again and we talked about what had happened between us. It gave a sense of closure of what had happened with our relationship and I was able to move on quite a bit since then. It also helped that I haven't seen him for the past 6 months. So yeah, I no longer have the desire to be with him anymore and he and I just keep in touch as friends (although sometimes he wants more but I refuse him of that).
But if you are convinced that you are in love with her, then fight for her. The chances may not be very high, but I believe that if you are truly in love, you would take that chance no matter how small. If you are having doubts due to the chances alone without even trying, then I don't think that she is truly worth it then.
Ultimately this is your call. I'm just offering some possible suggestions and solutions. It is up to you from this point on. Good luck.