once you've unlocked
head into the newly unlocked level, character and settings don't matter. all you can do is walk straight up into a flashy looking statue thing to bring up a dialogue box. choose "too easy" to get the mirror, which unlocks Inverse Mode.
if you instead chose "too hard", you get
instead, which unlocks Endless Mode, and Eudaimonia Machine will disappear from the level select. to get it back, just start a level with endless mode turned on, and quit out of it. Eudaimonia will reappear and you can go back in to choose the remaining option and get the mirror.