You have to time everything perfectly in order to pickpocket Daud and take his key as well. Follow these steps and you will be able to do it.
First blink onto the stairs at the right side of the room. Now blink on to the top of the bookcase by the stairs. This is where timing matters. Daud has a certain pattern to his actions. What you need to do is to blink right behind Daud when he is busy reading a book facing his table.
When you are behind Daud, you will be able to pickpocket him and take his key, which is hanging from the table. After you have done that, blink right behind the assassin who is at the exit and go out of that room.
All the actions described in the above paragraph should not take more than two seconds. Furthermore, your timing should be perfect to do it without any hitch.
After this you need to deal with the three guards in the next area. You need to take out one of them to proceed through this area undetected. The rest of the guards can be simply avoided. The assassin you need to take out is on the shade watching over the area. Take him out with a sleeping dart if you have one.
After that follow the white marker to the entrance of the tunnel. It will be in the same room as the window which you used to sneak into the base.
After all this, you are required to get through the quarantine wall and out of the flood district. In this area, stick to the left side of the buildings and use them as a cover to transverse through this whole area unseen. Use blink to get to the gate lever. Pull the lever and go through the now opened gate.
Climb on to the chain and then use the blink ability twice to get to the fuel cell. Pop it out to deactivate the wall of light and then use the service entrance to get out of the quarantine zone and into the Old Port district.
From there, you need to make your way back towards the Hounds Pit bar. Follow the white marker to the sewer entry.
If you do this quickly you can avoid the butt-load of weepers who will gather at the sewer entrance in a minute or so. Rotate the crank wheel to get into the sewers.
Follow the marker to the sewer entrance. There will come a junction in this area, where you can be killed by those stink bomb throwing pods
It is impossible to travel through this area on the ground level. However, if you look up you will see a pipe. Blink on to that pipe and it will take you straight to the sewer’s exit to end this mission.