If you already completed the "Enemy of my Enemy" Quest, you should have the Triumph for it under Triumphs-> Destinations-> Titan-> World Quest completed. And you need this Quest for the Rat King so if you have the Weapon or the Quest for it, you completed the "Enemy of my Enemy" Quest. If you did not complete it, Sloane should have it in her inventory. The step would be something with Fallen that you have to kill or something like that. About the Adventure: You actually don't need the Adventure. You can go to the room on Patrol and open the door.