Fiona Apple is EII - don't let the fact that both of them are singer-songwriters fool you
Anyway - I thought I might explain myself why I think she's not IEI. Purely coming from VI, she lacks that ethereal IEI look - to me, she looks much more like an ExE. What's more, she seems to be really working her way throughout her own experiences of pain, trauma, sadness and idleness. I find her much too 'scattered' to be that sweet INFp that everyone loves. If it helps - and I know some people don't like to type others through their work - her lyrics are mostly about personal experiences and stories, and a lot of the recurring themes revolve around personal transformation and catharsis. IEIs tend to write more about time itself, her focus is mainly on the tragedy of life. ExE from me.
I like how you think outside the box. It's funny because I relate to so much of Fiona Apple lyrically. At one point watching her in an interview also, I could have swore she was delta NF, despite all the supposed Fe demonstration (so I actually was like...could she actually be IEE???). I also agree that not all rational types and EII are super stable/completely introverted etc. as people like to imply. I considered EIE for Fiona too due to potential Si PoLr with all her health issues. There are so many shades to all of the typings and a plethora of historical information backing why someone's type may seem different than it's more classical description. Very cool. From what I know she seems IEI to me, but I don't know a lot.