I always pay mine in cash. It may be that you found her through a tutor agency? In which case they can take quite a chunk out of the money you give the tutor each time they have a lesson. That way they can let the agency know of some but not all the hours they teach.
I know mine has to pay 20% to the agency each lesson.
That, and by the time she has declared it (if she does) doesn't leave them with a lot, and if they are a good tutor I would just go with it if it is not a hassle for you.
Dont know about the returning of the cheques for last 2 weeks though, that seems like a bit of a palaver.