no no no you're not allowed to feel guilty! You can't afford to feel bad, you need to use your energy in a constructive way right now. She does not understand what this or possibly any job, actually means, and it was not your responsibility to teach her that. 6 weeks is a bit long IMO, make the next one 4?
You might have to look for some kind of temp nanny agency now, as she may just leave in a huff at any time once given notice. I can't advise you on that, I cut back on my work in the end myself in extremis (I sacked one by day 3, as very bad feeling, paid her for week and took key before she got her breath, got a lot of threats and harrassment later asking for more money, but she went away in the end) but a temp agency was the next thing on the list I planned to try...
Uwila I agree, if they aren't doing the job specified I would think you should be able to dismiss, but probably you have to do the verbal then written warning stuff...