I have a sony Viao pcg-71914L laptop that is deader than a doorknob even though the battery and the charger have been proven to be good. No lights on at all when the power button is pushed. No sign of life at all. Is there a fuse in it somewhere that could have failed?
Remove the battery and unplug the adapter from the laptop, hold the power button down for no less than 30 seconds. Reinstall the battery, plug the adapter into the laptop and try powering up again. If that fails then it's likely the power board has failed and requires replacement. Depending on the design it may entail a complete motherboard swap.
Vaio laptops come with inbuilt startup assist methods. There should be 3 buttons next to the power. Try pushing ASSIST, it should give direct access to the bios. It will also show if it is just the power button that is broken or something bigger.