@Zwero Slay the enemies using drawbridge trap at area 2, sometimes the quests not instantly updated the score (strange), so maybe refresh game page is needed :/ @bakkart agreed :3
@Zwero at the second part (courtyard) you have a drawbridge trap (to activate it just step on the "arrow-button", enemies on it (only ground enemies of course) fall down and die.
Kill 20 (doesn't matter in how many levels and how often you had to use the trap) this way and you get the quest.