Recently, when connecting an external monitor to my laptop, KDE turned off both displays, which sometimes even endured when disconnecting the monitor. Resetting the display configuration on the command line helped: Before plugging in the video cable, I opened a terminal and typed the following command:

xrandr --auto

After both screens had become black, I pressed Enter and the desktop re-appeared on the laptop screen and I could continue configuring with the regular Display Configuration:

kcmshell4 display