In my limited experience of teaching my one and only child, the important thing is to have "zero emotional buy-in". You must treat her as if she's a stranger's child. You wouldn't yell at some else's child right? And you must let go of all thoughts that her progress somehow reflects on you as a parent. You mustn't compare her to any other kids progress either. Nor must you set expectations as to how fast she's going to get through the method book. Its much better to introduce her to a wide range of simple pieces from many sources, rather than racing through the method in any case.
Of course your child is not going to treat you with the respect that she treats a stranger and sometimes our sessions get pretty tempetuous!

Playing with her elbows and other crazy things, just to prove a point. But if its musical, or she's exploring the keyboard then that's OK with me. Tomorrow will be OK for the proper stuff.

Concentrate on her attitude to learning, rather than the progress she makes at the piano. If people ask I don't say "my daughter is a wonderful pianist" i say "she has a wonderful attitude to learning" and hopefully B leads to A.
Try not to let the whole lesson be corrections. This is very hard! One strategy is, if she say, stuffed up the rhythm yesterday, tell her what you expect before she plays and then don't worry about correcting notes and fingering until tomorrow. If she doesn't "get it" you need to break the problem down. Get her to sing the piece first. Sing the piece all the time yourself, in the car, getting dinner etc. If there are no words, make them up! Get her to play one note and count the rhythm, so she doesn't have to worry about reading and fingering while she figures that out. Clap the rhythm etc. etc.
Try to notice the good things, even if they are not perfect. "Your fingering was much better today". "your tone was very nice". "That was pretty". etc.
Ah its a fascinating exercise teaching your own daughter. A learning experience for both of us.
