Hi gminpa,
Thank you for the trouble shooting steps tried.
I suggest that you shut the system down, remove the battery and disconnect the ac adapter, press and hold the power button for 20 to 30 seconds. Now connect only the ac adapter and wait for 30 seconds and turn on the system. This will drain any residual charge in the capacitors.
Let me know if the power light comes on and also the status of the Caps lock, Num Lock and Scroll lock keys on the keyboard. In case the power light is on and if Caps lock, Num Lock and Scroll lock keys light flash, it indicates the system completed Post. Connect an external monitor to check if there is display on the external monitor.
Revert with the results to assist you further.