I don't think your daughter is without a father, I think she doesn't realize yet how lucky she is to have one whose heart aches for her.
She's 16, who knows what she is thinking, I'd bet there are times even she doesn't.
I'm going to put just a small portion of my life and experience with my own child out there for you.
My son is now 23, during his childhood he was bombarded every other weekend with how terrible I was, a liar, a *itch, how we were not a family because of me, etc, etc not just me but my entire family. We were all bad, horrid people.
I know this because I heard it with my own ears, no speculation involved.
And, yes, there was lots of alcohol involved.
The person who thought this was OK was his own father, he did this to pay me back for leaving him (his own words)
This entire time my family and I never said a bad word, we simply showed my son who WE were, we're good people.
My child missed part of what being a child is all about, simple enjoyment, he was angry, bitter and no amount of counseling could get to the root of his reasoning for being this way.
(And he had one of the best child psychologists in town working with him)
Just about the age that your daughter is now my sons' eyes began to open, slowly over a period of about three years, he got it.
Without going into more details that you don't need, I'll just say that my son and I have a good relationship.
It was a long, hard road dealing with all of this, when all you really want to do is shake your kid and tell him who did what and to snap out of it.
I'm not telling you my story (condensed version) to give you false hope, because no doubt in some aspects your child's view of reality is a bit skewed.
I hope not to the point that there is no turning it around for her.
You can't give up on her, I don't think you ever would.
At this time in her life the best you can do for her is to show her the person that you ARE.
I don't mean to push her by any means, it's going to take some patience.
I really do hope for your sake and hers that things turn around, she's becoming a young lady and young ladies need a good man in their corner.