Unfortunately I have had lots of experience with this sort of behaviour from my SIL in the past. Combination of an opinionated, dominating personality and unexpected aggressive outbursts caused by her drinking. I always got the impression she thought she was better than the rest of us, could do no wrong and was entitled to give unwanted criticism when she felt like it. The last straw was when she set up a secret meeting between my estranged mother and my daughter and was bullying my child to go in and see 'grandma' just don't tell your mother! I did protest to my brother about it but was told it was his house and he could do what he liked. Shortly after that my SIL cut off contact with me and I haven't spoken to her for 10 years.
I realised, after a lot of therapy, that she was just another element of a highly dysfunctional, enmeshed family with a complete lack of boundaries. I tolerated a lot of abuse from my DB and SIL over the years. Lots of not turning up to dinners and events I organised, no warning or apology. Lots of snide remarks and criticisms from SIL. I later discovered she was nasty on the sly to my daughter. They basically walked over me because I allowed myself to be used as a doormat, a hangover from my childhood. My DH, now ex, didn't support me or stand up for me.
Cutting me off was the best possible solution to my SIL problem and I felt an immediate relief from not walking on eggshells around this ridiculous, puffed up creature. Ironically she has been in touch recently because they have now gone NC with my mother and now consider aunts, uncles, cousins etc to be important! Never going there again!
You haven't elaborated on the backstory to your SIL problem but consider this, what is the dynamic in your husband's family? Is he the family scapegoat that can do not right? Does your SIL feel entitled to criticise and put you down? If this is so, you have little chance of changing the dynamic. The lack of remorse and inability to apologize is a big fat indicator of a rigid, nasty attitude towards others.
My SIL wrote me an email slating my awful DM (wholeheartedly agree with her) but never apologized for her own behaviour towards me. It's like nothing happened and we can just meet up after 10 years and play happy families. Completely nuts!!!