I am looking for effective ways to permanently kill black mold in a
small bathroom.    The over the counter bathroom cleaners will clean
it, but will not kill it effectively.     The over the counter methods
also make me quite dizzy even with the ventilation that I can get in
that room.

I would recommend you visit http://www.black-mold.com/index.html for information

I had black mold in my bathroom. After a lot of research on the
subject, here's what I did. I sprayed all the mold down with X-14.
This is basically a strong bleach, the only thing that really kills
the stuff. If you don't have good ventilation in the bathroom, buy a
fan to move air around, and wear a mask. Let it sit for a while before
wiping it off. If wood is infected, wiping rubbing alcohol on it
works, too. To keep mold from coming back, after the walls are clean,
paint them with mold-resistant paint. Invest in a small heater (the
kind that sits on the floor is fine), and run it every day after the
family takes their showers. The mold should stay away.

Kriswrite

You could try a bleach pen (tube of concentrated bleach gel).  You can
paint the bleach over the moldy areas, let it sit/dry for a day, and
then wipe it clean.   Because it isn't an aerosol, the fumes aren't
too bad.

I heard a NPR report on molds and the expert said that one of the best
ways to kill mold is to use Borax.  Borax is a soap that can penetrate
porus surfaces.  Other cleaners just skim the surface.  Here is a link
to this particular product:

http://www.dialcorp.com/index.cfm?page_id=55