I think she's an INFJ 4w3...or possibly ENFJ, though I'd say to me she comes off more like an introvert when she's with someone like Graham Norton who is a Ne dom for sure.
My main Ni suspicion is the way she describes her songwriting. She describes her songs as "girls" that just come to her, some in parts, some ready to record. She describes herself as a kind of a scribe.
There's another curious fact I read about her making her setlists. She makes a new one for every City she plays. The interesting thing in this is
how she goes about deciding what to play that night: she and her assistant get together in the morning over breakfast and go through the local newspapers for a few hours and then she says gets a feel for what needs to be played that night.
She's pretty hands on, producing her own records, being very much in charge of every aspect of her career from selecting her road crew, to keeping herself very much involved and knowledgeable with the business side of things. She has a home studio and keeps everything to a small intimate crew... I can't really put my finger on it and what it is that exactly makes me think INFJ, but it's my guess.
I read Piece by Piece, a book where she's interviewed by Ann Powers and a little bit later Just Kids by Patti Smith who in my mind is highly likely an INFP. Patti Smith and her contrasted...I think Patti Smith was more a born bohemian, she just thought "I am a writer" and never really had a plan per se. She went to New York and did various things to express herself artistically and things kind of fell in place as if by accident, whereas Tori Amos had a specific goal and very consciously worked toward that.
..There's also that whole Y Kant Tori Read episode where she kind of just went with what people wanted of her and what was popular at the time to break through and only really begun finding her own voice in her mid twenties. I kind of think a Fi dom would just be themselves so to speak and not so much care what the world wanted. I kind of think it's possibly harder to balance your own needs and those of others with Fe in your make up ....but I'm not sure. It's hard to distinguish between INFs.
...I'm of course very biased as she is my all time favorite songwriter. Just thought it's fair to mention that.