Thanks for all the advice.
We hired her on our own, not through an agency.
Our contract says the 1st 4 weeks are a trial period - we are just past that, but to be honest I don't think I specified what the notice period would be after that. I was quite ill for the first few weeks she was here so didn't really have an opportunity to observe her "in action". I can't say I have had discussions with her along the lines of change-this-behaviour-or-you're-sacked - I don't believe in threats like that, and in my view if someone needs that level of warning then they're really not going to be a suitable au pair for our family.
This is not that kind of situation. It's more that, because I am at home relatively little for 4 days a week, and DH almost not at all, we need someone who is a much better communicator and either more experienced or more of a natural at childcare (and mess-tidying, homework help etc) than this girl ever will be.
I would certainly give a week's pay in lieu, and pay for a ticket home if that's what she wants to do. I am also planning to reimburse her for the English course, whether she stays in our town or goes home. We paid for half of it anyway, but I am happy to pay her back for the other half of it.
I don't think it's realistic to wait for her to find another family. We are not in London - there are not millions of families needing an au pair here, she can't drive so wouldn't be able to work in one of the surrounding villages which limits her options even more, and her English is really not great at all, so I am guessing it would be challenging for her to find another family. My feeling is she might be happier working in a more adult setting anyway, but she has to make that decision herself. I think she might suit a family where the parent was around a lot more than I am, to provide guidance, and perhaps where the children were older and required less interaction - but again it could be quite a while before such a family turns up.
We also have alternative (much better) childcare sorted with a person who can start right away - unexpectedly our previous au pair said she would be able to come back for a few months, which would work far, far better for us (note this did not happen until after we had already decided the current one had to go). So from my perspective and that of doing the best thing for the children, it makes sense to say goodbye to the current one sooner rather than later.
I am not so concerned about "are we doing the right thing" from a legal perspective - I will make sure we are, as far as the financial side. It's more about what would be a humane way to tell her, and what sort of help/approach would be appropriate after that. I am not too keen on the idea of her continuing to look after the children after we tell her we're letting her go (just as I would not ask someone to work out their notice if I fired them from an office job where I work). But obviously it is complicated as she is living here. I guess maybe I can talk openly to her (to the extent this is possible!) about whether she could stay with a friend or something. Or pay for a hostel for a week or something if she wanted to stay and look for work?