I paid for a 3 day professional conference in March that was to be held in July. On the day I was to leave home, I became quite ill and was unable to attend. The conference registrar is refusing to give me a refund of my registration because "we held a spot for you". It cost me $450. How can I persuade them to refund my money?
You can't. I'd put a fair amount of money that the T&Cs for the conference include details of how much of your fee you get back if you cancel at certain points before the conference, and it will be "nothing on the day of the conference or later" - if you even told them you were cancelling, which it sounds like you might not have done.
And this is the right policy for them to have. Their marginal gain from you not showing up is just about zero, but they couldn't resell your spot to another potential attendee at such short notice.
Both of the above answers are correct (+1). The conference spent the money on all of your training materials, facilities, speakers, A/V equipment, meals, and other expenses, and that did not change with your not attending.
Further cancelling the day before the event did not allow them enough time to go to their potential wait list to see if another candidate previously not going could then go.