This is not a job I could ever tolerate, OP. I cannot believe you have lasted this long. Start looking for another position immediately - your employer is never going to change. And NEVER agree to anything like this ever, ever again. You accept a job with set hours (7:30 to 5:00, Monday through Friday - or whatever) and have them guaranteed. If for some reason they do not need you for those exact hours one week, they are still obligated to pay you for those exact guaranteed hours. Those are set in stone.

Wow.
She just "laughed" when you mentioned you needed to know what time you got off at....??!

What a show of disrespect!

I would have a serious talk w/her in a very direct manner letting her know just how important it is to you to know your hours ahead of time.

She should understand that this is a totally reasonable request on your end & try very hard to rectify the issue.
Hopefully she will since you are an excellent Nanny that would be tough to replace.

If she still continues w/the current pattern, only you can decide if it is worth it to stay.

There are jobs like this, but they usually pay a huuuuuuuge premium for the flexibility and on-call nature of the job.

Only you know if this job is worth it to you. If I were young and unattached, and it paid really, really well, I could see giving over a year or two of my life to this craziness to save up good money.

If their paying just regular rate that you could get elsewhere? No brainer -- start looking.

Nanny working full time hours but schedule varies week by week. No matter how many conversations I have with my boss (the mom) I never get a schedule for the week. I kid you not up until a month ago there was an entire week where I had to ask her every night before leaving "ok, so what time do you want me to come in tomorrow?" and I also want to know what time I get off (so I can have some sort of control over plans in my private life!) and she laughs at me. Sometimes she even says sorry can we talk about it tomorrow-my brain is in a fog, I am so tired etc etc.
For the past 2 or 3 weeks I have been making my own schedule (I know most of the kids activities) print them on Sundays and give to her to approve it or adjust it. It still has not gone smoothly. It never goes accordingly but I am always flexible to adjust an hour or 2 if I can and trying to make a point for bigger adjustments for instance, last Thursday night she asks if I could come in on Saturday work the day and do a sleepover because she wanted to celebrate her weeding anniversary with her husband. I said no because I had plans but I find myself thinking I would have said no regardless because I think she should get her shit together.
How do I solve this without quitting? Sit down and talk to her AGAIN?? Suck it up and take it???