Instructions
Obtain the multi-tool. Enter the Clockwork Cantina and approach the gear puzzle. Solve the puzzle by placing the three blue gears in the middle of the board and one brown gear on each end. After completing the puzzle, turn the crank on the side of the board to earn the multi-tool.
Hop onto the garbage can that's located to the right of the gear puzzle. This will result in you earning the steam battery.
Exit the Clockwork Cantina. Enter the Steamworks Island Museum and approach the yellow machine located to the right of the museum's entrance.
Pull the machine's blue lever. This will extend the arm sticking out of the machine and provide you with an access point from which to jump onto the elevated platforms.
Run across the elevated platforms until reaching a display case. Take the dirty breaker out of the case, jump down to the floor and exit the museum.
Re-enter the Clockwork Cantina. Jump onto the large spring to elevate yourself to the location of a large blue lever. After pulling the lever, a ramp will appear.
Walk across the ramp until finding the steam chutes. Jump into the steam chutes to be shot up into the air. You will land on the roof of a dome-shaped building. Open the skylight with your multi-tool. After entering the building through the skylight, you will be standing in the mayor's office.