First thing you need to do is to stop thinking she's doing things to get a reaction out of you. She's just living same as you should be. Don't let her action or lack of control how you live your life.
"Should I tell me W that my mother has acknowledge that she owes her an appology??"
No. You don't really know the reasons why your mom thinks she owes your W an apology. It could be just because you keep demanding it of her. You can't force your mom to acknowledge to your W she was wrong the same way you can't force your W to get back to you.
You planted the seeds, so see what happens.
M-43 W-40 2D - 9 and 5
Emotion, yet peace. Ignorance, yet knowledge. Passion, yet serenity. Chaos, yet harmony. Death, yet a new life.