You receive this quest automatically, after you have completed The Man Who Waits main quest. After you exit Sanitarium (M20,5) you will be accosted by a Dunryd Row Messenger. Listen to the message and head towards the Hadret House (M20,4). Witness the meeting with Thaosem near the house and enter. Here, you need to speak to Lady Webb (she is staying in the room that used to be inaccessible). Tell her about the events in the Sanitarium. Apart from that, she will ask you to report to her on the activities of the Leaden Key Brotherhood.
Return to Lady Webb after you have disabled or destroyed the Engwithan machine atop the Tower of the Heritage Hill (M24,6). You visited that location in the Undying Heritage main quest. At the same time, in front of the entrance to the Hadret House, you should meet a Courier, who gives you a rare pistol.
Return to Lady Webb again, after you have completed the Through Death's Gate main quest, i.e. after you have examined another Engwithan Machine in the ruins of Cliaban Rilag (M33,7). From Lady Webb, you learn that you need to obtain an invitation to the animancy hearings.
To obtain the invitation, you first need to win the favor with one of the three factions in the Defiance Bay -House Doemenel , The Knights of the Crucible or the Dozens. House Doemenel is a big manor in the center of the Brackenbury district (M20,3), The Knights of the Crucible are stationed in the keep, in the First Fires district (M22,1), whereas you can find the Dozens un the Admeth's Den Copperlane district (M18,5). To learn more about the individual factions and the quests connected with them, see the Defiance Bay - Choosing the faction chapter. After you have obtained some reputation points, with one of the factions (you are free to choose - this will not have any effect on the rest of the quest), meet the leader of the faction, at the faction's HQ and ask for the invitation.
IMPORTANT NOTE - Do not continue this quest, until you have completed all of the side quests in Defiance Bay. After this quest ends, you will not be able to return to the Bay, for some time. The same goes for all the rest of the important activities, such as exploration of available locations (it is imperative that you explore the Sanitarium, because it becomes inaccessible), or meeting with traders.
After you receive the invitation, from one of the factions, return to Lady Webb and talk to her. Travel to First Fires and enter the Ducal Palace (M22,4). After you enter, do not use the main door, because it is locked. Instead, find the access to the side balcony. Wait for the hearings to start (it is chaired by Duke Aevar). It does not matter, which dialogue options you choose, during the hearings. Regardless of what you do, the Duke is murdered towards the end of the scene.
After you see all the cutscenes and receive a vision, exit the Ducal Palace. You need to return to Lady Webb, i.e. to Brackenbury (M20,4). Since the city is being ravaged by fires, you cannot use all of the passages, as a result of riots so, take the Southern exit from the First Fires, i.e. the one right next to the Crucible Keep. Enter the Hadret House and go to the room on the upper floor. Touch the soul on Lady Webb's bed. Automatically, you leave the city and are promoted to the next act.