Make sure to explore a room located to your right before you head towards the airship. There's a Voxophone #52 next to Fink's body.
The room found to the left contains a table and there's an Infusion #18 there.
Once you've searched the office open the door Elizabeth has already used and start moving towards the First Lady. When you're on board of the airship knock on the door located at the back to call the girl and then interact with the ship's control panel to set course. Use the panel again as soon as things will start to complicate.