Once you go through the door, it's time to kill lots of enemies (see bug below). Eathe should be the one to do the finishing blow if not, the count of kills will not go up. Killing them with the poison skill will not work either. Once you've killed enough of them, talk to Eathe again.
She will be pleased at the killing and will give you another key.
If Eathe's quest is the last of the 3 you started in Galette, then once you complete the quest, it is possible to start her next quest almost immediately without having to travel back to town. After poisoning the well and speaking to her, Eathe will run towards the end of the wooden bridge. If you run up to her to speak to her again, you can start the next quest 'Into The Breach'.
Tip: Just kill them with longbow with a lots of poison feat (or a crafted one)
If the player exits back into the tunnels before the water has been sufficiently poisoned, the quest may become permanently bugged and not move on to its next phase. Leaving this quest unfinished may also prevent the player from fast-traveling from that point on. Luckily there is a bugfix available (PC-only). Check the talkpage for more info and a link.
Since more and more enemies show up until the water is poisoned, it is possible to continuously kill them (so long as Eathe does not land the finishing blow) without any real limits. However too many bodies may cause the game to crash.