So she is a passive-aggressive bully? If you have not read it, check out "Crucial Conversations." It is a book mostly for business/work, but it has good strategies to deal with passive aggressive people - the one that I like and use is threefold: 1) take everything at face value and 2) let the person know in detail that you don't respond to passive aggressive, 3) walk away unaffected. Passive aggressive people hate this since they want to be the teacher and you the student and you effectively turn the tables on them with this approach. After a while, they will passive aggressively bully somebody else. Most of the time, they will respect you more.
Get your wife on board too.
Example: Someone passive aggressively says "I guess that I will just do the dishes by myself!" I directly say "That sounds like a good idea that you had, and you should do that" taking their statement at face value. Then, I immediately follow up with "If you want help, then ask for help directly because I don't try and decode passive aggressive statements since the world of direct communication is oftentimes complicated enough." Then smile and walk away unaffected and like you are now the teacher and they are the student (roles got switched).
2 things about Mary Poppins that allowed her to get away with more garbage.
1). She was hot
2). She had a British accent
Does your MIL have any of these attributes?