Slate himself can be found at the end of path, close to an entrance to a new display. Before you take the vigor him you will have to make a choice what to do with him - you can kill the captain or spare his life. The consequences of this choice are minor. No matter how you've chosen collect the Shock Jockey vigor and start going back to the Soldier's Field.
On your way to the exit you should expect several more encounters with the remains of Slate's forces, including a single Raven. Once you've dealt with all enemy units use the newly acquired Shock Jockey vigor on an electrical node found next to a closed gate.