I believe there is a short somewhere in backlight circuit. When you turn on your headlights
it appears that those other modules that also have backlight (like climate panel) join the
speedo backlight and are controlled using the normal dimmer button. But if there is a
short then lights just won't work. It is also possible that the relay that switches lights on
is defective.
If you ran out of warranty then I would start with disconnecting 12V battery for few minutes.
Maybe it just went crazy.