You need to finish the mission without causing any guests to panic or guards to give alarm.
Some spoiler free suggestions:
The alarm can be disabled. The city watch officer in the foyer has the key.
A man has a proposal regarding Lady Boyle. Seek him out, convince her to go to the meeting place and ensure his plan comes to fruition.
Alternately, kill the target when you are both unobserved.
Finishing the duel in the garden will not disrupt the party.