Medium phillips screw driver. Small flat head screw driver.
My brights have never worked on my S13, and it drove me nuts. I started out with testing out all of the headlight relays (by the driver's side fender). They all seemed to be working. I had brand new headlights in, and since both the high-beams weren't working, it left me with the switch. I called Auto Zone to price a new switch, and it was $60+. Me being me, I decided to attempt to fix it. It was actually easier than what you think.
The first thing you will need to do is get it out of the car. Remove the 2 plastic pieces on your steering column. (5-6 screws underneath and then it pops apart.)
Then, you will need to unplug the 2 plugs leading into your headlight/turn signal switch. (it's worth a shot to try unplugging the plugs and replugging them in to see if they just weren't seated correctly. This may save you some work.)
After that, remove the 3 screws (2 holding the switch, 1 holding a wire to the switch).
Now, you can safely remove the switch from the car and take it somewhere well lit with room to work. I chose my kitchen table...
After the switch is out, you'll need to remove the actual switch from the shell. This part is extremely tedious, and my lack of patience nearly caused me to break it. You'll notice there 6 snaps around the sides holding it together. Using a very small flat head screw driver, you'll need to work them apart and pull the switch out.
Once the switch is out, you'll have this:
You'll need to remove the plastic cover. Break out the flat head screw driver again and start working it off. You'll notice the little +s to bend it past the notches.
Now you are where the problem seems to be... You'll notice the 5 "prongs" inside. Lift up on the ends of them and use your little screw driver to scratch the surface on the contact points. (Don't forget to clean up the prongs as well.) Don't bend them up too far. If you crease them, they're ruined. It seems they build up corrosion and this is what makes you have a "bad" switch.
After you are done with that part, you still have a couple more connectors that it would be wise to clean up...
I'm sure this part already fell off
Since it just lays on top of
You'll notice in the right side of the picture, there's a connector there. Treat it like the contact points from before and use your flat head to remove any corrosion/gunk stuck to it.
Once you're done, lay the flat piece back on top of it to make it look like this...
MAKE SURE YOU LINE THE LITTLE SQUARE UP.
Once you have all of your contact points cleaned up, put everything back together like it was before, and place it back in the car. Voila, it should work. Make sure your signals and everything else work as well. If not, take it back apart and figure out what you didn't put together right. If the lights don't seem to come on at all now, you screwed up the little white square piece that just lays on it. If you want to test it before you put it in the car, when you lay the piece on top of the other, try turning your headlight knob and see if it moves for every click.
Hopefully, this will save a few of you some money. I'm sorry I don't have more/better pictures. I was simply doing this for myself to see if I could fix it when I decided that it wouldn't be a bad idea to take pictures in case it worked. Sure enough, it did.