Dear friend,
Here are some important truths to keep with you always as you seek to put your life back together.
" If we confess our sins, he is faithful and just and will forgive us our sins and purify us from all unrighteousness." (1 John 1:9) You have taken the important first step toward redemption. Try to build on that as you go forward.
God loves you unconditionally and stands ready to answer your prayers that are in alignment with His will. Since He hates divorce (see Jesus' pronouncements in several of the gospels), it would seem that putting your marriage back together would be something which would be in agreement with His will.
It's not going to be easy. As much as He loved us (enough to die in our place and suffer the wrath of His Father), Christ warned us, "I have told you these things, so that in me you may have peace. In this world you will have trouble. But take heart! I have overcome the world.” (John 16:33, emphasis added) But He has promised to be with you every step of the way; "I will never leave you nor forsake you." (Joshua 1:5)
If I were you, I would totally humble myself before my wife, and God. Ask for her forgiveness just as you asked for God's, but accept the fact that she might not be as forgiving, at first, as God has been. Tell her how much you love her, and how you regret the things which you did because of your addiction to alcohol, and how you have vowed that they will not happen again.
And most of all, "walk the walk". Treat her with the utmost kindness and respect. Show her the new side of your character in both word and deed. And give her some space to work things out in her own way, in her own time. I believe that you will get nowhere by pressuring her to take you back until she has seen that you are, indeed, a new person in Christ.
And don't stop praying. I truly believe that God wants to see your marriage healed. But you must be patient and let Him work out the details. Just trust that His solution will be the best for all concerned.
May God answer your prayers, and may you find peace in knowing that Jesus walks by your side.