It can either be,
"Why does she write bookS?" or,
"Why is she writing a book?".
The second option you gave is correct, but why do you want to change singular to plural in the first option? The original sentence has one, singular, book.
I did that on purpose because "why does she write a book" is unnatural speech. It works with plural normally: "Why does she write books?" It also works normally with negation: "Why doesn't she write a book?"
The correct structure with a singular book is "Why IS she WRITING a book?" This is because, "She writes a book," is unnatural. It should be, "She is writing a book."
I'd say "Why is she writing a book?" sounds better. "Why does she write a book?" sounds like an odd construction even though it's not ungrammatical.
I put "why does she write the book?" and it wasn't accepted. How would you say that in Polish? Thank you!
It's just an oversight, added now.
Although perhaps a better option would be "tę książkę", literally "this book".