A fixed external supply will run a strip forever, because it is not monitored and have overcurrent protection that are incorporated in the TV's design. It may be that that one faulty LED is the cause of your problem. LEDs can go short as well as open. In your external supply, allow 3V per LED in the strip, eg a strip of 10 LEDs needs 30V supply. It could be that the TV uses 6V (5.6V) LEDs which changes the whole equation.
Do you have a LED tester? Each strip should give the same reading (volts). If one strip is out of kilter with the others, it is suspect.
These testers are not conclusive because they're designed to limit the current in the strip to around 30mA, unlike in a normal operational TV where they use around 300mA, so they are driven much harder and a defective LED is more likely to crack.
My education into the foibles of LED strips seems to be never ending. What happens if you turn the tv on without the display and diffuser sheets. Provided everything else is hooked up, you can operate the LEDs without the display. LEDs will still operate and you may spot another issue with the LEDs. If there are joiners at the centre of each strip, these sometimes have connectivity problems and may have to be bridged out by soldering jumpers across the loins.