My WS and my sister were really good friends, even before I met him. She was devastated when she found out about the A. Because they were friends, she felt a betrayed by him as well.
I believe wifehad5 gave great advice. My WS confronted the issue head on with my sister. He couldn't do that face to face because we are far away; instead, he called her and spoke to her at length about everything. He took the time to apologize to her for hurting her and our family with his actions, as well as hurting me. He asked for her to give him time to make amends. He did not ask for forgiveness, only for her to keep an open heart.
I know that was a really hard phone call for him to make, but it went a long way to healing their relationship. It will never be quite the same, but it is again friendly, loving and strong.
It will be really tough, but will show your BW and your SIL that you are willing to face the consequences of your actions in order to reconcile. Good luck.