If the blue led lights and does not flash it seems to indicate everything is OK. Check the video cabling and the monitor.
But given the fact that the fan stays on after a reboot on a cold system, you could be right, it might be dead. Call Intel support and let them decide.
The cable and monitor work fine with another computer (like they did with the NUC until yesterday). So the problem is indeed the NUC.
your problem might be the same as in this thread: After Windows 10 upgrade monitor with MiniDisplayPort not working properly If this is the case your NUC is basically fine but the screen isn't recognized anymore. I have the same OS and NUC and had the same problem. To make sure it's the same problem just test your NUC with a different monitor and connect it over hdmi.
Thanks for your input. I am using HDMI, so this is not the issue.