Northener, what does the letter say? I assume it is you stating your case and trying to get her to see that she has caused this problem? Personally I wouldn't care about her using it against me but I think she is using your good nature to play more mid games with you and that she is making you run after her and I wouldn't give her the satisfaction.
As you know I am in a similar situation but a few months down the line and all I can say is that my mum has behaved exactly the same. I have written several emails to her, trying to open up the channels of communication at first, telling her how hurt I feel because of her behaviour and in the end telling her a few home truths - and all she has done is tell me she will get in touch when she is ready to (like your MIL - making sure it is on her terms) and she hasn't done. What she has done instead is show my letters to my brothers and sisters and get them on her side against me.
I don't regret anything I have said because I have come to the end of my time anyway with the game the family are all playing (with her pulling the strings), the manipulative way she behaves, the badmouthing me and the way she gets others in the family to think badly of me. But if I had walked away 3 months ago without sending her anything it would have had the same effect - ie got me nowhere.
As someone said on my thread, she is NEVER going to listen to what you have to say, admit she has behaved badly and that all this was her fault and come running back saying she doesn't want to lose you. So spare yourself the emotional energy that this is using up and walk away with dignity.
This is her fault, not yours and we all know that, your SIL probably knows it and even your MIL probably knows it deep down. But banging on the door trying to get her to see it is not going to work. If she was capable of being reasonable and direct with people about her feeings she wouldn't be the manipulative gossip that she is. You have blown your MIL's game of divide and rule apart (she is probably jealous of your relationship with SIL) and now you need to get off the playing field and use your energy up where it is appreciated - on your lovely DH and children.
If you are interested my thread is called "Would you cut your mother out of your life - long sorry" and it is long!