If the black wire in your picture has been located to the mounting stud by you, and originally was inside the rubber housing, then your bulb socket is no longer grounded. Remove the rubber grommet and resolder the black wire correctly.
To confirm operation, simply touch the black wire as it is directly to the battery NEG and the Red wire to battery POS. If it lights, you're good. If not, you know the black wire is a problem as wired/shown.