An idea for a novel

I've had an idea for a novel floating around in my head for a number of years now. I want to throw the idea out there to see what you all think about it.

A woman is murdered, and blame is assigned to a notorious serial killer. One person who knows for sure that they didn't kill this woman is, of course, the serial killer themself. Rather pissed off at having a copy cat, the serial killer becomes a detective to track down the real killer.

The novel will be told as an epistolary story, as the serial killer has been engaged in a long-standing email exchange with the FBI agent heading up the investigation to track the killer down. We only see the killer's side of conversation, unless they quote the agent's email. The serial killer details their investigation, how they infiltrate the murdered woman's social circles in order to figure out whodunnit.

Although it's something that the FBI didn't know yet, it becomes apparent over the course of the novel that the serial killer is a woman. Or may be a woman. Of course, a serial killer corresponding with the law may be the definition of an unreliable narrator.

Thoughts?