Presumably she's looking at it on the basis of being with you 24 hrs/day, so wants to be paid on that basis? Which is really not, in my experience, the norm.
We took our nanny skiing and paid: travel costs; ordinary salary for what in the event were shorter hours, just like everyone else here; accommodation, meals, expenses; ski lessons/ski hire (because she was doing lessons with the children).
I think your nanny may be working off the tales of bad experiences that fly around in the nanny community (and do of course happen sometimes) - where a nanny really is expected to be permanently on duty - sleep in the same room as children, do all the getting up at night, plus working long hours without breaks. But you clearly aren't planning anything like that!
I think on "where to go", the first step is to say you've asked around and fully understand the risk she might see of you taking advantage, but you absolutely won't (so v v clear on hours, duties, etc.; and agree an express overtime rate for anything more which you ask her to do); and if she is still adamant that the norm is 24-hr pay, then I'm afraid you're at risk of her taking advantage - throughout the job - and you might want to decided stopping now. (Presumably you've got a "trial period" built into your arrangements?)
You might also want to have a look at the chat board on the nannyjob website, as this issue comes up again and again and again, and you will then see some of the horror stories she might be worrying about. Plus you will also get yet more confirmations of the norms ...