Keeps switching itself off (completely off; not even on standby).

You need to do some in-depth troubleshooting, starting at the power supply module.
Please read my guides first.
If you are going to DIY and have proper tools, electronics knowledge, and know safety precaution then please read on:

Basic LCD monitor and TV troubleshooting guide:
http://www.fixya.com/support/r5093881-lcd_flat_panel_tv_troubleshooting_guide
http://www.fixya.com/support/r6150077-basic_lcd_monitors_troubleshooting
http://www.fixya.com/support/r7406380-tv_training_manuals
Failed TV and Monitors pictures: http://s807.photobucket.com/albums/yy352/budm
Learn about bad caps: http://en.wikipedia.org/wiki/Capacitor_plague
Parts: www.shopjimmy.com
Parts: http://interliquidparts.com/

Capacitors kit: http://lcdalternatives.auctivacommerce.com/Default.aspx he can make you a set of caps.
Or www.digikey.com, just make sure to use caps with low ESR, 105c, high ripple current and long life rating such as PANASONIC FM or FC series.
Please post back what you find.

Try change the tv with another power a/c cord,see if that solved the shutdown problems.If not.The tv power supply board have intermittent shorts in it some where in the tv power supply board.Tries websites like Shopjimmy.com,Ebay.com to buy a whole new refurbish power supply board for the replacement.