The primary adjustment for the brightness should be in the monitor itself. OS X includes a tool to help you adjust the settings, by showing you patterns with instructions about how to adjust the monitor's settings. You are also prompted to adjust the output from your Mac at the appropriate time. If you make adjustments from the wrong place first (by not using the tool) you may get a less pleasant, effective, and more fatiguing image. To use the tool, go to System Preferences > DIsplays > Color > Calibrate.
The main problem proved to be that the USB cable from the monitor to the Mac Mini.
There was no brightness slider on the Displays page.
I re-plugged the USB a few times to improve the connection and I got the brightness slider back on the Displays page. I then did the calibration as you suggest.
I can now stop squinting at the super bright monitor. What a relief.