Suddenly can't access Mac Mini
While you're unable to see what's on the screen at present, you'd still be able to use the keyboard to reset the NVRAM and SMC, so I'd try doing that to see if it has any impact.

Reset the NVRAM

Your Mac mini uses Non-Volatile Random-Access Memory (NVRAM) to store various settings. You can reset this as follows:


Shut down your Mac mini. 
Press the power button and then press the commandoptionpr keys. You have to make sure you press these keys before the gray screen appears or it won’t work. (In your case, since you may not get to see the grey screen, all you need to do is press the keys immediately after powering up)
Hold those keys down until your Mac reboots again (i.e. a 2nd time) and you here the startup chime.
Let go of the keys and let your Mac reboot normally.


Note: When you log back in you may need to readjust some of your system preferences (e.g. speaker volume, screen resolution, startup disk selection, time zone information, etc).

Resetting the SMC on a Mac mini

Your Mac mini's System Management Controller (SMC) can be reset by following these steps:


Shut down your Mac mini
Unplug it from power
Wait 15 seconds
Plug the power cord back in
Wait for an additional 5 seconds
Press the power button to turn it back on


Once you've reset the NVRAM and/or SMC, let me know how you go.