It is good that he admitted the "I'm an alcoholic when I drink all day". It means he does think about it and he knows how it affects you and the kids. You are right that it may be a while before he really does do something about it but he IS moving in the right direction now. This is huge step for him!
My sister told me that the hardest thing that she ever did was to make that first admission that she is an alcoholic. It was a huge first step for her. It's like finally opening the door and once the door is open then they can start to do something about it. For her it was easier to go to NA (Narcotics Anonymous) meetings than to go to AA. It was easy for her to admit addiction... but alcohol? OMG if I'm an alcoholic I'm just like my father... like I said, very big step this was for her.
I believe the prayers for your husband are being answered, the light in him will shine brightly soon I just know it will.
Oh and don't forget to follow my half-a-honey-bun diet... It does really work.