Mamafox - Reconciliation is a gift offered by the betrayed. It is not something that you, the betrayer, go and request. You may be scum, but she forgives you and reconciles. Or you may be the perfect, saintly sister-in-law and person from now on, but she wants NC with you forever. It is her prerogative.
Meanwhile, you need to figure out why you cheated so that the next time you have low self esteem and your wonderful boss/coworker/stranger at Starbucks attempts to seduce you, you will not succumb.