Kai is standing on the balcony overlooking Nariko's battle. Walk around this room until you come to a door with an action point by it. Trigger that action point.
Once you talk to the guard who refuses to open the door, several heavily armored guards will appear. These are the ones you have to shoot in the head.
Explosives line the walls of this room. You can shoot your arrows through some flame and into these explosives to detonate a nearby guard.
However you want to do it, kill all of these guards.
There are a number of beams crossing the center of the ceiling that Kai can run across. These beams allow you to keep a safe distance from the guards.
When all of the guards are dead a gate will open. Head out the gate and across the walkway. At the end of the walkway you will find a room. Spy on the guy in that room.
Now walk back to the previous room.
Aftertouch an arrow through some fire and then guide it over the walkway and through the small hole in the door.
The entire west wing will blow up. Once you've roasted the guard, head back to the room to catch the captain to get the password.
Now that you have the password, go to the other door and try the action button one more time.