Whenever possible, improve your relationships with Nyx (in house hades) and Chaos (in chaos land) by talking and giving them nectars. Eventually they will mention something about reuniting. Progress the story by going back and forth to talk to them (takes forever). Seems like equipping the chaos egg as your keepsake forces a chaos gate to spawn in the first few rooms, but i might be wrong.
Nyx will tell you that the only way to reunite them is to repair the Eldest Sigil. That is the red hole in the back of the administrative chamber, which you will also need access to, to finish this prophecy. To repair it, you need to pay the House Contractor 3142 darkness. The work order is on the first tab.
Once it's repaired, talk to Nyx again. She'll be happy you did it, then she will disappear from the house on subsequent visits. Go into a chaos gate and she should be standing near the boon. Talk to her to fulfill the prophecy. Go back to your desk to complete it and you should get the achievement.