IAT, I think all you can do is remember In-laws are humans, just with anger or sadness for what you did to their son.
I'm VERY lucky, I still have great relationship with MIL, she is a very hands on grandmother to our children, loves to visit after work, we visit her, kids stay over. We spoke after she found out. She told me she'd hurt me if she ever found out i'd hurt her son again (Justified!) We later spoke a bit more about it. She held me at arms length for the first few months, but then we found we were talking like normal, hanging out with the kids like normal and the awkwardness went away. I think she knew how sorry I was. She knew I accepted my blame, and how much I hated what I did.
Since my partner has ended things, and moved out, I have seen her a few times. We are still on absolutely great terms. I told her I will love her son for the rest of my life, but I need to let him be happy and be away from me. That i regret everything i did, and wish i can turn the clock back. That i know I can't so all I can do is let him be happy. Let him heal. But even so I will wait for him for the rest of my life. She cried, hugged me, and told me I can talk to her, if I need it. That she was sad for our split and wishes she could help him through the pain more. I said I wished that for her, and him too.
Anyway I'm getting to the point. If they can see you are truly sorry for your actions, that you are putting in work to help BH heal from your betrayal, that you aren't glad for the family isn't what it once was, focus on being open, honest and communicative, talk to them like they are still family, act with humility, take an interest in their projects, things going on with their lives (they're still people too!) it will get easier over time.