You guys are lucky. It didn't unlock for me until I had collected every bandage (including the leftover bandage in Cotton Alley) and had beaten every level in every world except for 7-20x, which is probably because I collected the bandage in that world.
As for the death count, it was 9,4xx. I wanted to finish on 9,001, but I overestimated my ability. I only had 6000 deaths by 94%, so I figured it wouldn't be too difficult to meet that goal. I was wrong