Hi there,
My Conrad Digi LS60 W soldering station also had a malfunctioning display. The soldering iron would heat up, but not all segments of the temperature display would light up. (Likely the same problem as the comment by BlueStyle above)
I managed to solve the problem by reheating the (soldering?) contact points between the display flatcable and the PCB of the soldering station. Now the display works again.
I used the tip of the soldering station to heat the display cable and pressed the cable firmly to the PCB using the tip of the soldering iron. A similar procedure can be found here: http://hackaday.com/2009/06/20/repair-a-malfunctioning-lcd/
Good luck with your repair!
Best regards,
Roel Wessels
The Netherlands