So if this woman was mentally ill/delusional, how did she manage to convince someone to pretend to be 'Jesse James' on the phone? Why did she calmly accept that the game was up when confronted? Admittedly I'm no expert, but I didn't get the impression that she was having a deeply held delusion challenged.

The whole story makes perfect sense if you assume that she simply enjoys the feeling of power she gets from tricking and manipulating people, with no regard for the consequences at all. If you're really determined to treat any form of unpleasant behaviour as an illness you could call her a sociopath. Either way, the author and his friends did exactly the right thing - get her away from her current victim, and warn as many people about her as possible.