The only way to enter the duke's manse is through the "Duchess in Distress" quest.
To do this, when you first gain entrance to the Duke's castle in Gran Soren, speak to Mirabelle who will be standing in the the first room of the castle or in the gardens. Do the "Arousing Suspicion" quest she gives you. Now continue with the main questline until you've finished the quest "Reward and Responsibility". Do not speak to Aldous again until you finish this next part. Find Mirabelle outside the castle by the stairs and she will give you a set of armour to grant you access to the manse so you can retrieve Aelinore. Follow the quest marker to the Blighted Manse and put on the armour to enter and unlock this achievement.