Found mistake in my professors book: Tell her or publish? If so, how?
First: Slow down.

In the academic world, books are simply collections of known things presented in a unified fashion. In fact, in some disciplines, it's almost unheard of for someone to release something novel in book form. Additionally, while book chapters are reviewed by editors and (occasionally) peers, it's not the same peer-review setting as we have in journals. To that extent, people don't typically write papers rebutting an interpretation put forth in a book; they write papers rebutting the article(s) upon which the book chapter is founded. If a book chapter does indeed contain an error, then it'll just have to be fixed in the next revision, if there is one. As such, your concern about publication is probably far overstated.

Secondly: Are you sure you have found an error? Has anyone else reviewed your work to validate what you've seen? Are you sure that you fully understand the topic? If I was a betting man, I'd place my money on the professor who's been studying the field for years and has (presumably) thoroughly reviewed the literature, so much so that they just wrote a book on it, over first- or second-year graduate student, as a simple matter of statistics.

Thirdly: You should definitely just go talk to your professor and ask her, respectfully, to clarify what she wrote. If anything, she'll probably be appreciative that you're going through it in such detail. If you have found a real issue, you'll make her aware of it, and if you're lacking some background she can explain it.