The trick to fixing a short in a light fixture is figuring out exactly where the short occurs. You will need to unplug and take apart your fixture to find this, unless the damage is visible on the cord or plug. Look for burnt or bare spots on the wire insulation, as these will indicate the short. A sharp kink in the wire may also be the problem site, if the cord is older and the wire has broken inside the insulation. Once you find your short, replacing the damaged portion is required to fix the problem.
Cut the wire with the cutting edge of your wire strippers, removing the damaged spot plus at least 1/2 inch on either side of the damage.
Strip the insulation off the last 1/2 inch of each of your newly cut wire ends. If the new ends are too far apart to reach each other, you may need to add a short piece of wire between them. Use the same gauge (thickness) and type (copper or aluminum) as your existing wire and cut it a little longer than you think you need. Strip the ends of this wire of insulation.
Heat a soldering iron and touch the tip to a piece of wire solder to allow the solder to flow onto the iron tip. Heat one bare wire end with the iron and press the end of some soldering wire against it so the solder flows onto the bare wire. This is called "tinning" the wire.
Tin each bare wire end, including your added wire, if you needed one to span a gap.
Hold the two wire ends you want to connect so they are touching each other and then press the tip of the soldering iron against both at the same time. The heat will cause the solder you tinned them with to flow together, fastening them.
Repeat the soldering process for each connection you want to make.
Wrap the bare wires tightly with electrical tape once they have cooled off. Stretch the tape as your wrap it, and wrap at least 1 inch beyond the soldered connection in each direction. This replaces the insulation you removed from the wire ends. Do not allow any bare metal to show.