So drunk she can't even remember how to hold the brush right?

Looks like she's writing random nonsense as well. Although it's quite neat for somebody who's supposed to have terrible handwriting.

There is nothing wrong with how she's holding the brush. From the writing pose, incline the brush a bit, disengaging from the 2 supporting smaller fingers, and you get Aya's hand position.
If you don't what I'm saying, loop up how one writes with a calligraphy brush.

The book appears to say 裏風流東方春 but I'm not sure.