To unlock Poetic Justice achievement or trophy in Dishonored, you will have to neutralize all key targets using indirect means. While the achievement description explicitly says Indirect Means, but it doesn’t necessarily enforce it on all missions, somehow.
Assassinate him while he is trying to kill the Captain. Or chock him out when he enters the basement with the captain.
You can sold them to Slackjaw to serve as slaves. Talk to Slackjaw and complete a side mission her offers you. After doing that, he will take these twins as slaves.
You can knock her out and give her to her lover. You will find this man wandering on the first floor. Talk to him. Next, you just need to get Lady Boyle in Kitchen so that you can knock her out to kidnap her.
Go to the broadcast tower and talk to the broadcaster who will suggest getting Lord Regents tapes from his safe (935) and broadcast them over the air to dismantle his government. Doing that will not only complete the mission but will also unlock political suicide achievement.
Poetic Justice achievement will pop after completing Mission #6.