Let her go. The problem is that although you say you didn’t intend to cheat she has no way to verify that. Because that’s just what every caught cheater says. And even if you could proof that you didn’t meet anyone it looks like it wasn’t for a lack of trying, just a lack of opportunity.

The trust is broken. That’s irreversible. She realized that she can’t fully trust you again. There is nothing you can do. How would you even try to earn her trust back? Wasn’t that what you tried for 4 months? It didn’t work. Telling her that you were faithful since is not doing the trick because that was the baseline of her expectations. That’s why cheating is so terrible, in a monogamous relationship even if you can’t agree in anything the monogamy is the baseline of terms you agreed on. And that foundation was taken away.

Let her go. Learn and do better in the future