Have you tried a "direct boot to internet recovery"?
That is...
- power off
- press the power on button
- immediately hold down Command-OPTION-R
- you may need your wifi password
- the utilities take a while to load, be patient.
Does that work?
If you can't even get there, it may be time to take it to a brick-n-mortar Apple Store genius bar and let them have a look at it...