Already stated, but maybe not undestood…
The switch you are tapping into does not have 12v+ on one side and 12v- on the other side. You are probably tapping into 12v+ on one side and 12v+ on the other side. You need to tap one end (12v+ of the LED) into the switch wiring (after the switch if you want the switch to operate it) and the other end to a ground/12v- wire (which will be on the other side of the light bulb switch the switch original operates).
I guess electrically speaking you can tap into a 12v+ wire, interrupting it with your LED, and it becomes positive on one side and negative on the other, but having a light bulb on one side also interrupts the electric flow with what is basically an electrical resistor (the bulb filament) which will hinder the 12v flow of electricity and render the LED dark.
Remove the bulb by cutting it out and splicing the wires together where the bulb used to be and this should solve your issues, but if you want the bulb to remain then you will need to tap into the 12v+ side of the switch and then wire the LED to ground wherever you find ground, or on the other side of the original bulb, although for me the other side of the bulb just somehow prevented current from flowing to the bulb - preferring the LED route instead. Im over my head now so will yield to someone…