All of this really needs to have been laid out before you took her on- she is your employee, not a service provider (like a childminder). Every nanny role is different. For example, in my current role I do all of the children's washing but the cleaner does all the household ironing. In my previous role, I did everything. A nanny friend of mine, on a similar wage, does none of the laundry as they wanted her to focus on the children solely.
You are expecting way too much if you expect her to read your mind! Every role is so different there is no way of knowing what is expected of you. She has no way of knowing your financial situation, I've worked for families hat would be horrified if I took a packed lunch to soft play when you can buy (expensive) food there, and others who would be horrified if you bought all the food there!
As to the nanny thinking the lists don't apply to her- well if you've never told her that they do, then they don't! I've had jobs where roles within the house were very separate- cleaner changed all the beds, employer was unhappy if I did it as it took time from the children. And at the nursery I worked before I began nannying, a qualified member of staff doing the jobs that the unqualified assistants were supposed to do got you in trouble- I got called into the office once for doing the washing up because the assistant was busy! Very, very extreme but she may have encountered this kind of attitude in the past.
Have you considered the schedule? In my current role I sometimes struggle to get the playroom tidy for the parents as the schedule is so tight- I'm cooking while the children play, then feed them (Can't tidy then as the baby is weaning), then strait up stairs for bath then parents usually home while they're in the bath. So I end up tidying on my own time, while MB finishes off the bath.
Again with the driving, have you spelt it out that you prefer her to walk? In the past I interviewed with a family that preferred you to drive even very short distances as they didn't want to children outside too much- yes I turned that one down, I walk everywhere and kids need to be outside! It may be that she has gotten used to a particular way of doing things.
Of course I'm only speaking from the nanny's point of view, as that is where my experience is, it may well be that you laid out the roles in the contract and she is ignoring it in which case you need to have a meeting. Sitting her down at a time when the kids aren't there and having an honest chat. You may find she is unhappy with certain things but as it's been two months doing them, then she feels that's how it must be done. That's from experience! Nothing can change without communication.