One way to look at this is that you have dual responsibilities and accountabilities that are in conflict with each. To accomplish one, you have to abandon the other one. It is truly an "either/or" choice.
If you stay with your alcoholic wife, especially at the level of advanced alcoholism you are describing, you cannot create a healthy emotional and physical environment for your daughter.
If you take your daughter and create the life she deserves, your wife cannot be part of that life as she is now.
I am not Catholic, but I seem to remember - correct me if I am wrong - that when a child is born, if there is a medical problem, it is the child's life that is protected before the mother's life.
If you take your daughter and create a happy, healthy, sustaining environment for her, what you are really acknowledging is that your wife has abandoned you and your daughter, not the other way around. As you say so poignantly, "she's a broken shell of a beautiful, vibrant woman that I once loved."
You didn't leave her and break your vows; she left you for the mistress of alcohol. Your wife has the choice to give up her alcohol and become healthy.
Your daughter, at 13, is at a very crucial stage of her life where growing through adolescence into the woman she was meant to be will be challenging enough without the devastation of a disruptive dysfunctional mother. Better for your daughter to face the truth of what her mother is now than to feel that she is responsible and failing her mother. She needs all her and your strength to grow into who she has the potential to be.
Having raised 5 kids, 3 of them daughters, mothers and daughters go through a difficult relationship during their teen years. Your daughter needs to navigate the normal normal adolescent angst with her mother. To do this in the context of profound alcoholism, will amplify the difficulty exponentially. She needs to model herself on her mother at the same time she rebels against her mother and becomes her own person. Yet, as the child of an alcoholic father, I know that children, often without knowing it, think "if I were a better daughter, my mother wouldn't behave like this. It is my fault; I am guilty; I have failed." This just isn't going to be easy unless she internalizes that her mother is very ill, and that she is not responsible for her mother's illness or behavior.
Just my thoughts, take what you want and leave the rest.
Glad to have you aboard here on SoberRecovery. It has been a godsend for me for the past year. I left my abusive alcoholic husband of 20 years, and a year later, I am happier, more contented, and free again. There is joy in my life again.
I felt anguish at leaving someone I loved so deeply, but living with him was so despairing that I almost lost myself. It's not that I don't still love him - I probably always will at some level. I was going under for the third time, and I had to save myself since he would not give up his addictions to repair our life together.
Now, the light is back in my life and in my spirit. I wish the same for you and your daughter.
ShootingStar1