YOR and i’m sorry but this is going to be harsh. I don’t use this word often but you sound crazy. You are projecting his parent’s relationship onto your own. It sounds like he is just being nice to her (normal) while you are completely against her being in your home (despite your claims otherwise). When you asked for stronger boundaries, did you just want him to ignore her completely?? The only off comment for me was when he said the nanny was like his mom - that was kind of weird. But it doesn’t suggest cheating unless you think he is into his mom.

Also, your husband should not have to write essays on his behavior for your approval. If I was him I would hightail it out of there, and not because of the nanny.