I'd agree that your wife may already know or suspect.
My STBXAH still cannot see that he has problem and cannot admit that he is an alcoholic, so I can't say with any certainty what I, as a wife of an A, would have wanted out of the conversation if he had freely and honestly admitted he's an A. I am certain that, knowing what I know now about alcoholism, I would need time to digest the information. I would want to know what his recovery program is; how often is he going to AA; is he also seeing a counsellor: where he is finding the tools he will need to work on his recovery.
I think first and foremost though is I would want the time and... space... safety (that's not quite the right word....)... to think, feel and work through my emotions, and to feel like I could ask questions.