When you are faced with taking care of the Pendletons, you will have several options, You can kill them in one of many ways, sure, but you can also deal with them nonlethally by dealing with the gang at the distillery. When you visit the distillery, you'll immediately notice that it has a little problem with the zombified weepers. If you poisoned the still, there will be weepers locked in a cage to the side of the front door, and more weepers will appear after your meeting with Slackjaw. As you head in, grab the distillery reserve key in the same room as the Incendiary Paste schematic, and head outside to the front yard to open the gate and find a Bone Charm here.
Anyways, head down to the distillery and meet up with Slackjaw. As you go, find the Brewery area and locate teh valve wheel. Turn it to open up a shutter and then head past to find the Incandescent Paste blueprints. Head down afterwards to talk with Slackjaw, and negotiate a deal. Agree or disagree to his plan to take care of the Pendletons nonlethally. If you agree, the bulk of your work in the level is now done.
If you want to kill the Pendletons, you'll have your work cut out for you. Either way, you will have to tackle a group of tough guards and weepers as you leave the compound. Three of Daud's assassins will be waiting for you as you exit the distillery. They'll spot your quickly if you aren't careful; if you want to simply kill them, the guards by the distillery will help you out when they see the assassins teleporting. If you want to avoid them, try to hit them with sleeping darts or blink away from the situation before they see you.
As you leave, check the room above the distillery entrance occupied by one of the assassins to find a Bone Charm on the top level, in the storage closet.
Another optional objective will now present itself to you. You can pickpocket Griff, the guy you freed from the thugs earlier, and get some cool stuff. You can also buy plenty of supplies from him legitimately, better preparing you for the missions ahead. Just note that he charges more than Piero does back at the compound.
You may also wish to find Crowley, who will be dead in Dr. Galvani's home. Check the corpse on the third floor to discover him. Of course, you'll have to avoid the four City Watchmen on the middle floor and the top floor. Try using Blink to get up to the second floor balcony to sneak past the bulk of them. Inside of Dr. Galvani's place, you will find a Rune. It's in the good doctor's secret lab behind the bookshelf, sitting snugly where the rat used to be.
While you're knocking out optional objectives, be sure to loot the Artist's house. There will be two enemies in his home trying to crack the safe in the wall. Head through the second floor balcony to get the jump on them, choking them out or putting them to sleep from behind. Loot the thugs, grab as much goodies as you can from the house, and loot the safe with the combination of 879. There may be other combinations to this safe, though; others have been reported online, but the 879 code worked for us. You will find a Rune inside of the safe. Even if you break into the safe on your own, you need the verbal combination from the artist to complete Slackjaw's quest. You'll have to head into the Golden Cat and interrogate him to get the combination from his mouth. Also note that all three of the paintings in the Artist's house are Sokolov Paintings, and thus collectible.
You can also visit Granny Rags for a second time in this level. She will be in a new area, a "home" in the undeerstreet area filled with weepers. You can nab an Outsider relic here, but little else. Just nab the Bone Charm from under the dead end with the weepers to check that one off.
Along the way towards your real objective at the Golden Cat, you will come across a woman being harassed by two members of the City Watch. Take them out in some way or another to save her and get yourself a key to the Artist's house. This will also make your Chaos level go down a little bit, as long as you do it nonlethally.
So, it's now time to break into the Golden Cat. As usual, there are plenty of ways to do this. The sneakiest way is to go through the Captain's Chair Hotel, just like Slackjaw will tell you. The building is across from the Golden Cat, where you will find a Bone Charm as well sitting under the stairs leading from the first to the second floor. You can easily sneak past the city's guards in the hotel, and come out on the roof which will lead you via a series of catwalks and windowsills to the Golden Cat.
Also, note as you are heading towards the Golden Cat that a Rune lies in the guard shack to the right. Sneak into the Golden Cat via an open window on the second floor when you're done, unless you're crazy and want to break in through the heavily-guarded front door.
You can also sneak into the brothel from the back, if you know what we're saying. You can get here by running through Granny Rags' new digs, which will take you to the back door of the establishment. Watch out for any wandering guardsmen, and slip in through the door when nobody is looking.
Now that you're inside, you'll have to find Emily and deal with the Pendletons. The order in which you do these things doesn't really matter, and if you entered through the window on the second floor she will literally be right in front of you. Emily is in one of the small rooms on the third floor. Rescue her and tell her to head down to the door in the back. She will be safe down here for as long as you need to run about the brothel, so don't worry about following her just yet. Also note that one of the rooms upstairs contains a Rune.
Now that you've saved Emily, check out the locked room on the third floor to find a Bone Charm. You can easily grab this by shoving your blade through the prostitute inside's head, but you can choke her out or whatnot for a clean hands playthrough.
Now it's time to interrogate the Artist and get that combination. Find the Silver Room, aka the torture chabmer, on the lower levels of the brothel. As you go, look for a Rune in the second floor foyer across from the madam's office. A guard will be out front, but make a noise to distract him and the Rune is s good as yours. Also as you go, watch out for guards and girls. The girls will raise an alarm when they see you, meaning that you'll want to avoid them or take them out. When you get to the Silver Room, interrogate the Artist to complete Slackjaw's request.
Now it's time to kill the Pendletons. If you want to do so nonlethally, simply return to Slackjaw and he will take care of your little problem. Otherwise, you can kill both brothers personally. Head to the Gold Room at the top of the highest floor in the garden chamber area. Watch out for the guards, who will be trying to peep in on the action. Head out onto the balcony to get outside of the Golden Cat, and use Blink to get over to the balcony outside of the Pendleton brother's room. Burst in and take him out through whatever means you deem necessary.
Now it's time for the second brother. Head down to the Steam Room and take out the two guards outside. You can kill this brother by steaming him alive by using a valve in the maintenance room nearby, or you can simply run into the room from the outside balcony like before.
Now return to Emily at the bottom of the Golden Cat, and take her to Sam by the shore. Your mission is now complete.