Translation:She takes notes quickly.
"Fastly" seems like it would be reasonable, but it's not technically a word. "Fast" is both an adjective (describes nouns) and an irregular adverb (describes verbs, adjectives, clauses, etc but doesn't end in "ly"). So the correct way to write it would be: "She takes notes fast".
So I've been looking at this question for a while (it's been sitting at the back of my mind for around a month now), but I'm afraid that I will be unable to answer it in an understandable manner. Regardless, I will try my best here.
I think I have to start with clarifying what the particles we have here do. I will do this by breaking down the sentence thus:
かのじょは ~ speaking of her
ノートをとるのが ~ the action of taking notes is the thing that
はやいです ~ she does quickly
Putting it together: "speaking of her, the action of taking notes is the thing that she does quickly."
Placing the の particle after かのじょ would indicate possession (translating to "her note"). In this case, that's not what we want, because then the sentence would be about her notes and not her. We want it to be かのじょは, because she is the person doing the action. Saying かのじょのノートをとるのがはやいです would roughly mean "her notes are being taken quickly," which would be what you would say if, for example, someone was quickly stealing her notes.