|HDMI port may require Windows drivers so if there is a VGA port use that, even a DVI port might work.|
That said, the shutting down points to something completely different, probably hardware related. If you get a post screen on monitor this way try to access BIOS set up and make sure that it will run that way for a while, if not then there is no possibility of reinstalling anything.
A screen does not break without some serious impact, this could have damaged many possible hardware components, on and off the motherboard.
If this machine is more than 2 years old, it might pay o replace the machine entirely.
You have to be a little bit crazy to keep you from going insane.