Sorry I don't know the answer, but £10 for half an hour on a casual basis does sound reasonable. I expect a professional teacher who pays tax etc. would charge more. I guess if she's happy to do it then you are all onto a winner.
Just asked DH about this. He says professional musicians here charge anything between £20=£30 (sometimes more) for an hour. But he reckons a reasonable amount in your situation would be about £12-£15 for half an hour.
I know the best qualified teacher in our town charges just £5hr (yes honestly). She's got the longest string of 'letters' after her name, but does it because she enjoys it, doesn't need the money (is now comfortably retired from her pro-music days).
Some pro's charge up to £50 a hour though (depending on who they are )
I'm in Kent and pay £12 for 1/2 hour for DS3's piano lessons. His teacher is highly qualified and is a bona fide teacher (i.e. with teaching qualifications) rather than someone who can play the piano and is willing to teach, IYSWIM.
Au pair has studied with Prague Conservatoire but I'm not sure if she has any teaching quals as such. I think I'll stick with a tenner for now and if DD is still progressing and more importantly enjoying her lessons I may offer her more around X-mas. DH plays piano so we'll be able to tell if DD's actually learning anything!