The WTCPL uses the same soldering pencil as the WTCPT, which I inherited some time ago, and Weller (now part of Cooper) still makes new replacement parts and tips for these old workhorses, which I think is fantastic.
Anyway, you'll find the datasheet for your model here, including troubleshooting guide, assembly diagram and parts list:
Weller Tech Sheet - WTCPL Series
In my case the magnetic switch was still working (there's instructions on how to test each part) but the heater barrel was open. I replaced the barrel and the nut (the newer barrels are slightly larger in diameter, so you might as well get a barrel nut at the same time) and the iron works perfectly.
You can get these parts from Digikey:
EC234-ND - HEATER ASSY FOR A TC201T IRON
BA60-ND - ASSY BARREL NUT FOR PES51&50
SW60-ND - SWITCH ASSM FOR SOLD PENCIL (magnetic heater switch)
Some are listed as zero quantity, but they order them and I got mine in about 2 weeks. Other replacement parts are available, but if you have to replace too much (or the more expensive parts) on the iron you might as well buy a new iron. When mine broke, I fixed it and purchased a new one that was adjustable for $100. Now my old one is the backup, and I don't mind bringing it to electronics group meetings.
However, this is a worthwhile iron to fix if the cost is not too great. The new fancier models have more features, but you really can't beat how durable and usable these old wellers are. While you're at it, you might as well get another tip or two.
Not affiliated with digikey, but they are a reasonable online supplier if you are in North America
For thru-hole work (which it sounds like you are doing) I would go with an active
flux that is water soluble. You need to wash the residue off since it is corrosive
and can cause shorts. I have had a number of boards mysteriously come back to life after
a good cleaning.
For thru-hole parts that may get damaged by cleaning (LED lenses) or SMD parts
with heatsink pads I would use the no clean flux.
I have been using the flux pens from Kester. 2331-ZX for water soluble flux and
the 951 for no-clean. To clean the water soluble flux I use an "acid" brush with
the bristles clipped to around 0.5". I run the board under the hottest tap water
I get (probably around 140-160degF) and scrub with the brush. If I do a small
repair job I will clean with the isopropyl alcohol I have on my bench.
Here some of the whys --
Why you need flux -- to get a quality joint at a lower soldering temperature.
Using a liquid flux lets you spread the flux better over the two surfaces
to be soldered.
Why choose an aggressive flux -- an aggresive flux is more tolerant of soldering
temperatures. As Leon mentioned this would be a good candidate for a hobbyist.
I like the 2331-ZX for this. Since these leave ionic compounds you need to clean.
Why choose no-clean -- eliminates a process step and the issues associated with
the extra process step. For example for soldering XLamp LEDs I use a no-clean
flux rather than risk scratching the lens.
For more details take a look at http://www.finishing.com/Library/flux.html
and the Kester website.
Best Answer
Plumbing solder is generally kind of thick - more than 1/8 inch. Electronic solder is usually much thinner - less than 1/16 inch.
You can tell if solder has a core by melting some on a piece of paper - it will leave a greasy, maybe sticky spot if it has a core.
Thin solder that leaves a sticky spot when melted should be electronic solder.
As @EJP says, the problem is usually a break in the cable near the plug. Cut the old plug off maybe an inch from the plug, strip the wires on both sides of the cut. Use an ohmmeter to find out which wire goes to which part of the plug, and solder the wires on the other part of the cable (the part that goes to the headphone) the the correct places on the new plug.