JWN, you are doing amazingly well!
It sounds like your daughter really wants to rebuild your relationship but she is possibly, at the moment, feeling scared that if you relapse it will be even more painful for her. You are PROVING to her that you can be the mother that she wants and needs by not drinking, but for her own 'self-preservation' (for want of a better word)she will probably be holding back slightly for fear of being hurt further.
My FIL was a chronic alcoholic when my DH was in his teens and stopped and started drinking for many years. DH feels like he missed out on doing things with his dad because he felt that, to his father, drink was more important than him. DH used to think he could 'save' his dad and was bitterly disappointed, and felt guilty, when he realised that he couldn't.
When I met DH his dad was drinking again, but could still 'function', but within 6 months had gone from just drinking in the evenings to vodka for breakfast and it was pretty clear that the drink was killing him. We managed to get him sectioned under the mental health act and from there he was lucky enough to get a place in a specialist rehab centre which follows the AA 12 step programme. We were very cynical about whether, at the age of 56 following years of drinking, he could complete the programme (although not to his face, but again think this was 'self-preservation' on DH's part), but he took each day at a time and worked through the programme and came out of rehab sober- and a much nicer person!
He met a woman there who is a lot younger than him, but who was also a serious alcoholic, and we were concerned that they would go back to drinking together- we thought it was a recipe for disaster!
However, 6 years on they are still sober and are having a fantastic time together doing 'normal' things like decorating their lovely flat, walking their dog, going on holiday etc and we could not feel more proud of them both. They have both been on a long journey and admit that they are still taking it a day at a time, but for the past few years DH has had a proper relationship with his dad and our DD is able to know her grandad properly.
Contrast this story with that of my uncle who died 2 years ago yesterday, from septicemia caused by his own waste getting into the sores on his legs as he couldn't walk because of the toll drink had taken on his body. His 'friendly' local shop keeper brought him drink, and charged his Visa card, until the day before he died.
I guess what I'm trying to say is:
a) It's never to late to stop no matter how long a person has been drinking for
b) Each day without a drink is a big achievment
c) Your daughter (and your DH) will be proud of you again and you have taken the first step to being in control again and regaining their trust and respect.
d) You have a journey ahead of you and some days will be easier than others, BUT YOU CAN DO IT!!!
I am filled with respect and admiration for you, you are awesome!!!