Really sounds like a power supply issue, probably the electrolytic capacitors. Once the electrolytic caps in the power supply start to go bad they have all sorts of power issues, from taking along time to turning on, or never turning on or turning off randomly. If you are not familiar with capacitor failure in tv's you can do a lot of research on the internet. It's really not a hard fix if you are patient and handy with a screw driver and soldering iron. Heres a good example of what you would expect to see in the power supply if you took the back off.: https://www.google.com/search?q=bad+capacitors+on+lcd+hdtv&espv=210&es_sm=91&source=lnms&tbm=isch&sa=X&ei=p_meUpbfG874oASQmoL4CA&ved=0CAkQ_AUoAQ&biw=1440&bih=779
I would replace every single electrolytic capacitor in the power supply. Also if you use higher quality capacitors it will last longer than it did before too.
If you are not comfortable with removing them, you could look into buying a complete power board replacement on ebay or bring it into a repair shop. Hope this helps good luck.