Well unfortunately I think I have bad news for you...
What you are describing sounds like a bad LCD panel, which would mean a new tv for you because the part is more than the cost of a new set. However you could have a bad connection on an internal connector called the LVDS cable. To check this however would require the back to be removed by a professional and the connection checked. There is a small percentage that this is your problem. In a lot of cases with these LCD televisions we see the panels fail in the exact same scenario that you have given here. Turn the tv on and the picture is good for a few seconds or minutes and then it goes black. Also blinking on and off is common in a LCD failure, then also when they go bad all the way they will cause the set not to come on at all exactly like you are explaining.
There are a couple of other options that could be causing this also but none of which are going to be cost effective. The price on the new sets have come down so far that repairing these small ones is next to impossible. I have not see and do not know of any simple fix or cheap repair for your situation, I see hundreds of televisions every month and to my knowledge a quick fix is not out there.
I am sorry to bring you the bad news tonight, please let me know if there is anything else I can do for you.
If you would like any additional information or have more questions please don't hesitate to ask!
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