Wow...sounds like something has been going on for a while. I'm sorry to read about your sister not speaking to you because you won't leave him. It's so easy, soooo easy for those untouched by infidelity to say "leave him." When you are married and have children...it's not so easy to leave. There is reconciliation and forgiveness. You need to tell your sister you are in pain and her support will make it better, her ignoring you will increase your pain.
Please, make a doctor's appointment and get tested for STDs. Ask your doctor if he/she can recommend a counselor. Suggest marriage counseling to your husband.
In regards to your husband trying and you pushing him away, well...he'll just have to keep trying.
He has lost all rights to privacy. You get to see all his cell phones, his credit cards, every email address he has, any clubs or online places he goes, you get every password. He needs to check in with you frequently throughout the day - you track his phone, he sends you selfies. You get to call his work or cell and learn exactly where he is and who he is with. You count the money in his wallet in the morning, and you get to count that money in the evening and he needs to show you a receipt showing what he spent his money on.
You get all the details. And you keep asking about those details. Cheaters will minimize what they did...eventually the truth will come out. Liars can't keep lies straight.
You didn't do anything wrong.