To be honest, op does not sound like a good boss. Who does not check babys bag at pick up?

I know nannies who try and live like the families they work for. They live well outside what they can afford to so have to work 7 days a week to have a life they can't afford

Her off hours are absolutely none of your business.

Address the issues directly. As others have said it doesn't matter why the problems are happening, deal with the fact that they ARE happening.

To put it simply she isn't meeting the expectations of her job. Arrange a meeting where those problems are addressed. Even the chance that she wants to leave early to get to another job. The other job isn't your concern.. the fact that she wants to leave early is. Let her know that you will no longer be accommodating her requests for extra time off. If an appointment is scheduled (doctor, etc.) she needs to ask when it is scheduled and give you as much notice as possible.

It sounds like she has some personal time she can use. Fine. When that is gone, it's gone.

Let her know that if she continues to fail at meeting expectations, you may have to consider finding a replacement.

I think you said everything perfectly in your statement right here. I suggest you do the same thing and tell your nanny what you just told us.

Let her know that what she does in her off time is not your business + that you know you have no right to dictate whether or not she takes a job or not.

However, for the sake of her job w/you, just let her know that certain things are sliding + that you are addressing them now so hopefully she can work on correcting them before they become larger issues.

As for being a flexible employer, let her know that by all means you want to make her job as easy as possible, however if she needs to get off early due to holding down other jobs than you feel you do have a right in saying she shouldn't hold other jobs if they are going to negatively affect the one she has w/your family.