[Mikie]

OK - strip the wires about twice as long as you want the final splice to be - lay the bare wires together pointing in opposite directions so it looks like one wire with a bare spot in the middle - hold the bare section in the middle and twist one bare end around the other wire tightly until you've used the end up - do the same the in the other direction with the remaining stripped end - put the soldering gun/iron on the bare wire, touching the solder to the wire slightly away from the gun/iron until the solder starts to melt on the wire and gets sucked into the length of the splice. DO NOT put the solder on the gun, wait until the wire is hot enough to melt the solder.
Your old wire to the light probably has corrosion in it, if it looks black in/around the wire you need to spread the strands and scrape it with a knife blade or use sandpaper to try to remove as much of the corrosion as possible to get the solder to adhere, otherwise it will bead up and drop off without penetrating the strands.


Oh, another point - if you're using a test light you want to attach the clip to ground then touch the probe to the hot side of the switch, if it lights touch the probe to the other side of the switch and turn the switch on to see if it lights, if so your problem is either in the wiring from the switch to the light or the light itself.