You could just try changing capacitors, but really without trouble shooting what the problem is your just shooting in the dark, sometimes it is a simple component burned out and is obvious, or bad solder joints or dried out capacitor.
With TV's sometimes the HOT transistor goes bad (Horizontal Output Transistor), and will shut down power supply (over voltage to tube can cause dangerous radiation),
With TV,s you can usually get kits (HOT transistor and capacitor, sometime fly back transformer), these kit are for the common problem the TV may have.
But if you are not used to working with electronics and TV's, I suggest do not mess with it, there are some killer voltages, and you would most likely have to call the repair man or find yourself another TV any way.
Forget fixing the TV, get the Kiln working, then watch it on Sunday evening :lol:
Or fix your kiln recover some gold and buy another TV.