If you cut the end off the lighter adapter, you will see that there are really two wires there: the hot and the common (or ground). Typically these connectors are round, with an inside and outside to the plug. The outside should be the positive. You may need to check with a continuity tester which wire is which. Sometimes, one of the wires has a white stripe, that is the common.
Find a spare circuit in your fuse block that works off of the ignition (or accesory) and wire from there with a new inline fuse. Route the wires up the A-pillar and splice onto the plug for the detector. Run the common the same way, but tie it to a good ground under the dash. This assumes that you mount the detector on your visor.
There is no pleasure in having nothing to do; the fun is having lots to do and not doing it.