Getting to the tunnel underneath Daud's headquarters | Mission 7 - The Flooded District Dishonored Guide
Last update: 11 May 2016
New Main Quest: Use the tunnel under Daud's base
You can reach the tunnel in two different ways. The best idea is to take the same way you came here, to return to the lower level of the headquarters, so that you are back to the room where your starting point was. Use the key to unlock two, inaccessible by default, cabinets (there is gold inside) and the door shown in the above screenshot.
Another variant is to find a hole in the building's wall, thanks to which you will be able to go around the abovementioned hole. The downside of this approach, however, is that, if you haven't disposed of the assassins guarding the walls at the building's back, you will now have to eliminate or avoid them.
Regardless of your approach, you will end up in a small room with chain hanging loose. Leap towards the chain, and start to climb down. I recommend against sliding all the way down, to the bottom. Instead, explore the building's floors as you go down. This will enable you to find a Sokolov's painting, a rune and a Letter to the Director.
After you reach Daud's underground headquarters.