Lisalisa - thank you for explaining the difficulties so clearly. If it is at all helpful, I can suggest some wording for the exit speech? - as I do think other posters have a point in at least giving her something to try to understand (if she is going to try ... she may not be capable of that, I realise, but I think as an employer you do have to give her the benefit of the doubt there).
Dejags' suggestion of "differing parenting skills" is an ideal starting point.
I would say - "it really does seem to us that we have fundamentally different outlooks on how to look after children. We try not too expect more of them than we know they can manage [if she challenges this you can refer to the eggs incident]; and if there is ever a problem, we do our best to talk about it then and there only, and then move on, leaving the problem behind us [if she challenges this you can refer to the "never taking shopping again" remark]. I realise you may feel that this is not teaching a child what it needs to learn - but it is our way of parenting, and I'm sure you will see that we just do not see things the same way. So, I am sorry, but we need to end this arrangement now."
I think you will find that she is not unhappy to leave.
If parallel personal experience helps, I had to terminate a nanny arrangement after a week - and she did say "But WHY?" in a shocked voice, and I had marshalled my thoughts and so it was simple to say, "Well, you were an hour late your first day; now I realise that can happen to anyone as a one-off; but then, when the following day we had agreed you would take ds swimming, you turned up without your swimming things - and it was then clear to us that you were not taking the job the way we had expected you to." It was hard, but having the words ready was necessary.