I'm going to approach this a bit like a cheating on your SO issue. I know that it seems hyperbolic to compare this to cheating, but the reality is that it involves a similar fundamental issue - namely how to regain trust after it has been (deservedly) lost.

The best you can do is both apologize and offer complete transparency. It's also good to go to a marriage counselor at this point.

For the apology: apologizing once may very well not be enough. You might need to apologize for this until you're absolutely sick of saying I'm sorry.

As for complete transparency - that means letting her know about every aspect of your life on demand. She wants to look at your phone history and emails? You let her. She wants to get a copy of your pay stub and monthly bank statements, so that she can check that you're not opening up other secret accounts? She gets a copy. She calls you when you go out to check up on you? You answer the phone, and you don't complain. In effect, you offer to completely sacrifice your privacy to try to fix the relationship. Whether or not this is absolutely necessary to save your relationship is mostly up to your wife.

How long you'll do both of those things can vary - there is a lot that your wife would need to do on her end in order to rebuild her trust in you. In fact, she would have to do most of the heavy lifting. You have to both convince her that you're worth it, and then make it as easy as possible for her to do it.

That said, I understand that the above seems extremely onerous, possibly even unfair. Maybe it won't be so bad, she'll be able to trust you again without it or quickly enough that it doesn't get so bad. After all, she may not think that it's as bad as cheating (but really, lying to somebody for years like this is pretty damn bad). If you really can't accept doing something like that, and it proves necessary, then your marriage is over.