The only way you can make your wife believe you is to....stop....drinking. If you told her again you will quit drinking completely for her, now you need to stop drinking completely, or it's another broken promise, one she likely expects you to break. Like most of us, you've put your wife through a lot, more collateral damage, somebody that doesn't deserve it.
It's time to accept you DO have a problem with alcohol, and possibly a big part of your wife's depression. It isn't easy to hear. Her road will be difficult for her too. Time, and proving you can be sober, is the only thing that will help you now.