The Gear is one of the objects required to be placed in the pedestal room in order to fix the Ink Machine.
Exit the pedestal room and continue along the hall going to the left. Take the first short hall on the left and make another left toward the Ink Machine location, where you’ll need to jump over an ink pipe on the ground. This is the same area where you had to place in the two Dry Cells, and as I mentioned in that post, if you look in the chest to the left, there’s a Gear inside.
To get back to the pedestal room, take a right at the first hallway, then another right. The pedestal room is on the right side.
