Now that you're in the party and you've completed the main side missions in the area, you're faced with the question of how to take out Lady Boyle, and how to discover which of the three ladies is the right one. That is, of course, if you don't simply decide to murder all three of them, a decision that will likely lead to lots of bloodshed but hypothetically is still totally possible. You'll just have to have the parrying and shooting skills to lay waste to all of the guards nearby.
If you're aiming for a more satisfying resolution, you'll have to find a way to get upstairs and figure out who the Boyles are. First things first, speak with all of the party guests. You will find a guest in a fly mask who has some particular knowledge of the Boyles and their costumes, but she won't spill until you head back to the main dining hall and bring her a drink in one of the glasses on the table. This info can be helpful, but isn't necessary to complete the mission. All of the mission critical stuff is located upstairs. There are a few ways to get up there without being seen. You can head up through the servant stairs, or steal the whale oil closet key from the guard near the Wall of Light to power it down before heading up the public stairway. As you reach the second floor, search the art gallery to find a Sokolov Painting, as well as a few other nifty items worth a pretty penny in coins. There are four guards on the second floor, so be careful and keep count as you explore. Keep crouched, and take the guards out one by one, whether it's by choking them out or by more deadly means. As you search, don't neglect to comb over the environment for more clues as to the identity of the Boyle sisters. Loot every guard you take out to get the most important piece; one of the guards will be in possession of a key that allows you to get into Dunwall Tower.
As you make your way around the second floor, keep an eye out for a second Sokolov Painting. One of the bedrooms on the second floor will contain the painting, which is of Hiram Burrows; you can use Dark Vision Level 2 to find the painting more easily. This is important, because right under the painting you will find some notes pertaining to the Ladies Boyle. Read it to gather some intel. Now, for the optimal kill, use Blink to get onto the upper balconies outside of the second floor. This will lead you up to the third floor, where you will encounter another guard. Take him out silently. Now you have free reign of the third floor, where each of the Boyles lives. Break into their rooms and figure out which is which by reading their letters. Their secret desires and weaknesses will be revealed to you, marking your target's location on your HUD and giving you a satisfying way to take out your target. Follow the waypoint marker to your target to initiate the takedown, and play along until it's time to kill them.
A non-lethal method is also available for this mission. In the ballroom, you will find Lord Brisby, who will tell you that he knows about your mission. He will ask you to meet with him. If you want to get through this without actually killing anyone, it'd be in your best interest to accept that meeting offer. Follow Brisby to his study and listen to his list of tasks. You'll have to knock out the target and bring her to Brisby, thus removing her from the picture for you and your interested parties and satisfying Brisby.
If you want to be a jerk to your hosts, note that you can also find a guest book on the first floor and sign it with your name, letting them know exactly who perpetrated this crime. It doesn't have ny impact on the gameplay, but it's satisfying enough to let your enemies know you'll always be there watching.
Before you leave the area, you'll want to check out the cellar. You can get a basement vault key from the bedroom near the main stairway, on a dresser. You can also pull the red sashes in the bedrooms to open up an attic tunnel. In the basement vault, take out the heart to find a nearby Rune. It'll be in the wine cellar, which you can access via the servant's stairs.
Now that you've taken care of everything in the party and gotten rid of Lady Boyle one way or another, all you have to do is head out and meet up with Sam. Watch out for the tallboys, who will still be patrolling outside unless you managed to kill all of them earlier. Use the canal to hide from them. Sam will have to move back because of heavy enemy presence, so keep your head down and stay underwater as long as you can as you head towards your new objective. You'll find Sam past the lock in the canal.