Go up and talk to Cecilia at the start, listening to her explain everything. It seems Havelock killed everyone, except for her and Emily. You are inside the Abandoned Apartment, so go to the door that goes outside (grab the key next to the door if you need it) and you'll find the place crawling with guards and Tallboys.
There is a very easy way to get rid of everyone without much work on your end, which involves rescuing Piero and Sokolov. Wait at the doorway until the Tallboy passes, then go down the stairs and hide in the bushes. You are waiting for an opportunity to Blink to the left side of the street right below you.
When you get an opportunity, Blink to the left and then onto the sign above the bar. Next, you will want to get on top of the building just next to the pub, but note there's a guard up here, so when you get up there, kill him. Climb the stairs and you'll be right outside Corvo's room. It is best to wait outside until the guards inside stop talking, then sneak up behind the guards left and kill them.
For the optional mission to rescue Piero and Sokolov, you will need to head downstairs and go to Havelock's room. There is a guard patrolling the second floor, so keep an eye out for him and kill him if he happens to come your way. Once you get to Havelock's room, look in the bucket for some blueprints, so take them and head back upstairs.
There is a window that leads outside and to the tower where Emily was staying, which passes over the roof of Piero's workshop. Go to the tower and then use Blink to get inside of the workshop, via the balcony. Go to the bottom floor and talk to Piero and Sokolov, who mention that they have finished a new kind of Arc Pylon, but Piero needs some blueprints to get it functioning.
If you already got them, hand them to him and you have a choice between killing all enemies or putting them unconscious. Naturally, you should kill them all, but you need to do one more thing, which is put a whale oil tank in the machine. Go to the door where you entered, get an empty tank, fill it up, then use the vents and planks to get to the rooftop. Flip the switch to make the cover go away, then put the tank in the machine, which will turn on and fry everyone.
If you didn't get Emily's letter when you were in Corvo's room before, do so now. After that, you need to get into the tower, but Callista has it. Of course, she was killed by Havelock, but the good news is that it's still on her body, so go over to where the body bags are and check Callista's for the tower key. Get up to the tower and go through the door, then use the signal to call Samuel. Talk to Piero to stock up on anything you might need, then talk to Samuel to start the last mission.
