I think there are many areas where you could argue, your kids, your rules, so if MIL wanted to give your child sweeties and you didn't, for example. But sleep is another matter.
Rather than handling at arms length/on the phone, which was frankly just being mean because you were both very tired from being at work all day, and DC not going to sleep, it's a terrible way to deal with it.
It also depends on your particular arrangement. If MIL offered to look after her, and eventually you took her up on the offer, gratefully, then fine, but if you asked her to help, then I'm afraid you just have to deal with it.
I do not understand how people assume that their elder relatives shoudl give up so much of the time to help, unless they've offered willingly (no idea if this is the case for you of course).
I actually have paid for care in the past rather than take up MIL's offer, so that I could stipulate thigns like this. It's a very difficult area, you don't need to be undiscerningly grateful (if that's a word), but equally, you have to be gracious about it too.
If, as may the point, MIL is determinedly and significantly undermining your parenting, then that's a different thing altogether, but nothing you've said gives me that impression.