Well, you made the first step of admitting you have a problem.
If I were you, which I'm not, I would tell your wife. Why keep it a secret? Everyone knew I was a drunk even though I thought I was pulling the wool over their eyes.
Honesty with her would be a great start to sobriety. We're only as sick as our secrets.
Then, you could be free to address your drinking problem. With her support, hopefully.
I hope your marriage is strong. If you love and trust her, she probably feels the same about you. So maybe you think you've broken that trust. But this can be repaired.
This is just what I would do. I sincerely hope it works for you.