Have tried twice now to complete this quest. No NPC shows up in or near the courtyard once the dungeon is cleared. Not sure if the quest is bugged or if I have to get to the NPC Before clearing the instance... Will update as I make the attempt.
Edit: FINALLY figured it out! To complete this quest, speak to Elder Sage Wind-Yi who stands just to the left of the steps to the entrance to The Temple of The Jade Serpent instance (OUTside in the temple grounds - BEFORE entering the instance!) She offers you an option to enter the temple. This phases you into a phased version of the temple which is NOT a Dungeon run, but the Temple at peace, with NPC's to talk to, including Elder Sage Rain-Zhu who is the quest turn in npc. She's straight ahead of you on top of the steps to the left. Once turned in, don't forget to pick up The Jade Serpent quest from Fei on the right side of the same platform!