Try grounding that pin, connecting it to +5V through a largeish (say maybe 1k?) resistor, connecting it straight to +5V (or maybe +12V if those don't work and you're feeling daring) in that order. Hopefully one of those will get it to turn on if it's in working order. Of course, disconnect that line from the TV first.
I was just on the verge of trying a couple of those combinations..... but I decided to backtrack a little first since I knew it originally would flicker and sometimes come on... and I could just tell the inverter was not on at all now.
Well, the only thing I had done was change out one cap on the main pcb and two small caps on the inverter itself. The main pcb has several of the same type and value caps, so I decided to swap out my new cap for one of those....sure enough the thing fired up the backlight. So I'm sitting there scratching my head because I know I checked continuity and such after putting the new cap in.... so I swap them back around.... the thing fired up.
What the heck is going on here?? No idea what was happening to not let it come on before. (this cap was one of the originally bulged caps, the other two were on the inverter itself)
The original caps are Rubycon 25v 470uf 85degree, the one I put in is a Jamicon 25v 470uf 105degree.
So it's coming on fine now, but I noticed that a lot of components are getting quite hot. Not just hot, but too hot. Small heatsinks you don't want to leave your finger on too long hot. Surface mount processors and such getting unusually hot as well in my opinion. It all seems to be working fine, volume, adjustments, channels, etc..... but I didn't want to leave it running very long because of things getting hot.
Thinking it may be something to do with the inverter again, I unplugged the inverter and let it run a bit. (blank screen obviously)
Still have stuff on the main pcb getting hot. So that tells me it's nothing on the inverter now.
Voltages still seem to be in a normal range according to the power supply and voltage regulators I've found on it.
Wondering if there are some more damaged caps causing this. (?)
This is when I wished I had an ESR meter.
Given all the monitors I mess with you'd think I'd have one by now.

BTW - pin #5 had 3.1vDC when the backlight started working again.
pin #4 dropped down to about 4.66vDC from the 4.95v I had earlier on.
None of the pins seemed to vary when brightness or contrast changes were being made.
But then again, as mentioned by MonMotha, this just may be something I can't measure correctly at the moment. But wanted to note that a DC reading was there when the inverter is "working" so to speak.