Hi Guinea Pig,
Your DD is in much the same position I was when I was a child - a father who had no interest and a mother who was very honest and loving and supportive (I think your DD is very fortunate that she has you, btw).
Bit of background: I never met my father - he died when I was in my twenties before I felt ready to go and see him. But I did consider it when I was 'old enough' so to speak - I was 17/18. I had sent him a letter when I was 14 asking for a photo but he wrote back to my mother saying 'discourage further communication'. I actually had a session with a counsellor at my mother's request when I was 17/18 - she was happy for me to try to go and see him, but she wanted me to be as prepared as possible for the eventualities.
In the end, the one counselling session made me aware that I would not be able to cope with it if he shut the door in my face; I thought it through really carefully and I decided that I wanted to wait until I felt strong and mature enough. Hard to paraphrase such introspection without sounding trite, I hope you can see what I mean!
Anyway, I have told you all that because I wanted you to know that I have an understanding of how your DD may be feeling right now, and how painful it can be to feel unwanted by your father, even if you have the best mother in the world. And I think that the best thing you can do is to carry on as you are - finding out what she wants and basing all your decisions on what is best for her.
It may be that he doesn't respond at all. It may be that he needs thinking time. You have offered him the chance to have a wonderful gift - contact with his child. It's his choice whether he takes up that opportunity. But the effects of his decision will resound through your DD's life. You have written the letter. It's presumably reached him. I think you are absolutely right to be honest with your DD, and that means telling her the hard stuff as well.
My mum used to say to me that my father knew that he would be welcome to make contact if he wanted to. She would say that it was impossible for us to know what his reasons were for not wanting to know me, but that she could only assume he was scared of responsibility and couldn't cope with being a father. She said this with no bitterness - just a statement of fact. She would always say that it didn't make him a bad man, but that he was not a strong man. I grew up understanding that my father wasn't evil or a saint - he was an ordinary man who for whatever reason had not been able to cope with becoming a father.
I'd imagine that your DD has had and may have some very sad times as a result of this, but it's clear that you will always be there for her in every way.
If he doesn't reply to you, perhaps she will want to write to him herself and some point. I wouldn't suggest this, but if she asks, I would agree and let her do it.
I think the best thing you can do is to listen to what she wants, which is exactly what you are doing. She wants contact, so you have given him the opportunity to fulfil that request. But you cannot force him, and I don't think that going to the pub would be a good idea - you would be putting him on the spot and be unlikely to get a good response.
You have offered him an amazing chance - I hope he takes it. But if he doesn't, he cannot be made to do what you hope he will. All you can do is to help and guide your DD through her feelings, and enable her to try to understand and make some peace with the situation.
There was a thread earlier this year from someone who was trying (as an adult) to contact her biological father - it's not the same situation as your DD is in, but it might be interesting for you to read it at some point in terms of what may or may not be ahead for your DD. I'll try to find the link.