I appreciate you sharing this. I got way too agitated from arguing with this person who is obviously unwell. Extremely frustrating but I should have given up earlier.
Madge is on a mission from god. Never expect to change her course with reason, and just try to see the humor in her cognitive dissonance.
If you don't have a taste for that, silo her. Regrettably, she's incapable of growth and enjoys her hugbox.