Oh Nix... I wish I didn't, but I understand exactly how your SD feels.
My Mother wasn't/isn't an alcoholic but completely and utterly emotionally unavailable among other things. (Thank god for my Dad) I suffered with the abandonment and all the issues that manifested as a result. It is unbelievable to a healthy person to understand and to a 13 year old, it's beyond all that a mother should feel for her own flesh and blood daughter.
I can understand that SD wants some admission of her lying to her, but the lying to her represents a greater loss felt to your daughter. Poor child....I really highly recommend a professional for your SD. Yes, she is lucky to have you.