What can you do to make things better?
Well, you knew this was a problem spot before the M, you talked about it at length, you made an agreement, and you broke the agreement. Thus you've proven yourself untrustworthy.
Then you lied about it, reinforcing to her that you cannot be trusted.
No wonder she doesn't trust you.
She won't listen to you or discuss anything with you because she doesn't trust you. How can she build a foundation on quicksand? She thinks she knows where it is, but then it shifts and is not as it seemed. Trust is a BIG deal! Only after trust is restored (at least partially) can you begin working on other things like your need for sex. I know that's not a happy place for you to be AT ALL but your current situation is the result of the unfortunate choices you've made. It can be fixed, though.
You need to EARN her trust back. I'd begin by offering to install "spyware" on the computer so she knows EVERYWHERE you go and can see all your passwords, etc. Better yet, install it, tell her you've done so, show her how to use it, and ask
her what else you can do to earn her trust. Something like that would have made a big impression on me when my H proved to me, through his actions, that he was not trustworthy.
I also recommend that you read the information on this site about the
Love Bank,
Emotional Needs (ENs), and
Love Busters (LBs). You'll want to begin working hard on filling her ENs
without any LBs. When she feels more loving toward you, you can bring up the subject of your own ENs, perhaps get her to read some of the info on this site or an appropriate book (His Needs, Her Needs, for example), and make your new M a more fulfilling place for the both of you.
You're not a bad person, you've just made a couple of bad behavioral choices. I'm glad you are wanting to make things better.
You can do this.