Agree with the other comment. If you want him to help, just tell him directly. Guys are not dumb but they like to dumb their way out. They know they should or can be helping out but they will always think “if she wants my help she should tell me”.
So now i just tell my husband directly, if you don’t see me attending to her when she cries, you should auto because it means at that point of time, I don’t want to handle (usually in the MOTN/early morning cause I’m a very light and late sleeper).
For your mil, if you can’t depend on your husband, just go straight to your mil and confront her. Thats what I did last time when I’m staying with my mil. I’m not rude nor disrespectful, just wanna clarify which part about me is causing unhappiness to you (mil). If you want to badmouth about me, go ahead but at least don’t let it get to my ears. My mil is rather tamed now or at least I don’t hear anything bad from anyone about me.