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Can you tell me if you get any sound? Does the screen ever light up at all?
Okay, if you've checked the power supply outputs and all the expected and marked voltages are present, then you have a bad main board.
This is an LED, not a plasma. Replace the main: https://www.encompassparts.com/item/8353719/Toshiba/75022781/Pc_Board_Assy,pe0953a
as your first step.
Note that the cost of the part is significant though, and for a 2011 model, it is at/past its service life at this point anyway.
Fixing this issue will not extend the life of the tv significantly, as other parts will continue to fail, so consider this a temporary repair. You may get another year or two out of it.With that in mind, it may make more sense to invest in a replacement tv at this point, as you can get a new set with a fresh warranty in the $600 range these days. I can recommend some, if you prefer.