Only after solving both of the cases will you unlock the endgame sequence with the Countess. This walkthrough will show you how to lure the Countess into a direct confrontation and how to deal with her once the confrontation has begun.
After the final case is resolved, both Lauren and Joey will return to Lauren's apartment. Once there, you will need to switch to Joey and have him talk to Lauren. You will need to go through all of the dialogue options in order to discover the connection between the two cases. As soon as you reach the point in the conversation where you have to choose which person is the common link, choose Joseph Mitchell.
Once Lauren and Joey have concluded that Joseph Mitchell is the common link between the cases, you'll need to head over to the New Yorker to confront him. In this conversation with him you will have a lot of dialogue options, but you don't need to explore all of them. The first step is to confront him about his writing. Joey will help you by pointing out that the paper is blank and there is dust on the typewriter. Once you get a long list of possible options, select 'Threaten Police Action' and then 'Martyr Yourself'.
After returning to Lauren's apartment from your meeting with Joseph Mitchell at the New Yorker, you'll once again have to have Joey go talk to Lauren on the Balcony. Their conversation will end with a knock at Lauren's door. Have Lauren answer the door to let the Countess in.
Once the Countess is inside you'll need to exhaust all of the dialogue options until she offers you help. At this point in the dialogue you can offer to help her, or to accept or reject her help. The outcome is always the same thing.
There is only one thing you can do once the Countess is choking you: burn her with a lit cigarette. That is the only defence Lauren has left. Simply click on the lit cigarette in her inventory and then click on the Countess.
As soon as you burn the Countess, Lauren will collapse and you will have control of Joey. What you need to do is follow a very simple pattern. Have Joey talk to her, then move Joey towards the Balcony once. Only move him once, then talk to the Countess again. Keep alternating between moving and talking and don't stop the pattern until Lauren wakes up and control shifts back to her.
With control of Lauren all you have to do is walk over to the countess with Lauren and click on the Countess. Lauren will do the rest. Congratulations, you have solved both cases and defeated the Countess.
For information on the Steam Achievements, please see this page: Blackwell Unbound - Steam Achievements
You can also find this video on YouTube: