Hound Pits Pub
Move forward and talk to Cecilia to find out about the current situation. Follow her up the stairs and grab the keys off the hook on the wall next to the door. Open the door and go through it to find the pub, except the pub is now overrun by guards. There are three guards in the street in front, another on the roof to the left and a pair of tallboys patrolling around.
The easiest way to access the pub is to use slow time (or wait until nobody is looking), then blink across the street and up to the first ledge. From here, climb into the open window. Enter the door on the right of the room as you enter, and then enter the room immediately to your right in the hallway. Look in the bin here for a Blueprint . Make your way up the stairs to Corvo's room. There are four guys in here. Hop out the window and wait for their conversation to finish. Afterwards two of them will leave. Return to the room and put the remaining guards to sleep. Check out your desk for a letter from Emily.
Climb out of the window and head over to the roof of Piero's workshop. You'll now get an optional task to make contact with Piero and Sokolov. It's a fairly easy task and you'll get a Rune out of it and be able to re-stock/upgrade one last time with Piero, but if you are happy with your powers at this stage, it is not necessary to complete.
Optional: Save Sokolov and Piero & Information on Havelock
Drop down off the roof to the right and onto the vent here. Follow it down and enter the open door, grab the Rune (1/1) off the table here and head downstairs to chat with the pair of scientists.
Piero will want you to go grab a blueprint from the pub. Luckily I told you where to grab it on your way through the level. Hand it over and after the conversations ask that they set the arc pylon to set them all to sleep (if you are going for a zero kill run). Talk to Piero and buy any upgrades or ammo you require and then head upstairs and use the dispenser on the left to grab an empty whale oil canister, fill it on the right and take it back to the roof.
Here you'll find a series of three canisters on the roof. Pull the switch here to open them up and place the now full canister in the empty socket. Pull the switch again to close it up. After the scene, all the enemies on the map will have been put to sleep, so feel free to wander around the map and loot whatever you like. To complete an optional objective return to the bar to find a letter indicating some of Havelock's plans.
When you are ready to continue with the mission, return to the roof and head out to Emily's room in the tower. Here you will be reunited with Calista. Loot the room and then interact with the object in the window. This will summon Samuel. Exit the door opposite (key is on the bed) and blink down to the beach below. Talk with Samuel to complete the mission.