Beat Aava, the King's Pet, ascend the stairs leading away from where you entered her lair and light the Grand Cathedral bonfire located in the Grand Cathedral.
Ascend either of the winding staircases beyond it, open the doors, enter the room and walk to the platform in the middle. When given the prompt, speak to the disembodied voice. That's Alsanna the Silent Oracle, who will explain the backstory of the Ivory King, his knights and the land of Eleum Loyce.
When you've exhausted her dialogue, she'll ask you a question — a favor, really. Answer yes, and you'll be on your way to the next phase of the DLC.
Alsanna's actions don't just affect what's in front of you, but also the world you've already explored. The wind has stopped, and the ice has melted. If you ever lit a torch and walked out of a room only discover that the raging storm extinguished the flame, that's changed. If you saw a chest frozen in ice that you couldn't reach, you can reach it now. What was once frozen and indestructible is now free and open for exploration. It's time to backtrack.
Beyond the platform where Alsanna spoke to you, you can (but shouldn't yet) descend a staircase and enter a fog wall for the DLC's ultimate boss. Instead of traversing the mist, exit Alsanna's room the way you entered and, before heading down either staircase and back to the bonfire, open the chest to collect the Vessel Shield x 1, which grants bonuses to adaptability, dexterity, endurance, faith, intelligence, strength and vigor and can also deflect spells.
Stand on the staircase before the mist-shrouded wall beyond Alsanna's room and look beyond it. You'll see four chairs and a large knight standing by the closest one on the left. Those are each subtle hints about what you should do before you enter the mist and confront the Burnt Ivory King.
Return to the Grand Cathedral bonfire and warp to the Inner Wall bonfire.