A 13A 125v power cord, as long as the connector fits properly, is not your problem.
Additionally, the symptom for the bad electrolytic capacitor problem, that affected many computers and TVs built 3-6 years ago, was that they stopped turning on at all.
From the information you supplied, your problem seems to be related to heating and the power supply would be the first place to look (e.g. if it fails to produce the needed outputs (e.g. voltage and current), downstream circuits will not operate.
Fortunately, with a working knowledge in electronics and a multimeter, a power supply problem is fairly easy to diagnose, repair and/or replace.
If you have a friend, neighbor with this ability, upon failure, I would begin with checking the power supply output voltages. They will be marked on the printed circuit board near the connector. In general, there may be up to (4) required outputs (e.g. 5V always on needed for turn on sequence, then for operation....5V to power the logic circuits, 12V and/or 18V to power the audio/backlight circuits).
If one of the output voltages is not correct, you'll have to diagnose the problem down to a faulty component level (requires additional knowledge, skills) or purchase a new power supply board (about $50-60).
One other thought is that many older sets required a cooling fan, if so, is it working?
Another consideration is that the prices on newer/better models have fallen dramatically. For example, 32" HDTVs are readily available on sale for $250-$300, or even an upgrade in size for not much more.