Although you will hear and see both, there is a certain ambiguity in them. If you wish to remove the ambiguity you will have to say something like one of the following, depending on what message you wish to convey.
I don't know what to teach or * how exactly to teach.
I don't know how to teach, or * what exactly to teach.
In those two, you could move 'exactly' to the place I have marked with an asterisk
I don't exactly know what or how to teach.
In the third example, if you were to place 'exactly' after 'know', as in your original sentence #1, it is not clear which word(s) it modifies.
It's probably just personal feeling, but I would use 'precisely' rather than 'exactly'.
- For Teachers