Bleach doesn't "kill" mold, in the literal sense. It oxidizes it, burns it up. Industrial strength peroxide does the same thing.
I had a spot once that would grow mold quickly. Bleach in a spray bottle cleared it up but it came back in a few weeks. After the 3rd. time the bleach lifted the paint off the drywall, so I soaked the drywall good so that the bleach soaked in to the plaster. I let it dry for several weeks, then painted the spot with Kilz primer and regular latex paint. The mold never came back.
YMMV