Well, I opended it up completely. The reason for this behavior was that the isolation of the cables of the backlight somehow was removed (maybe too brittle), resulting in sparks, a bit of molten plastic and burned cables. The high voltage cable was completely burned off. With the help of a microscope, soldering iron (and a soldering expert, thanks Eduardo
, silicone and special insulating tape everything runs again. I assembled it a bit wrong, because there is now visible backlight leakage in my menu bar on the lower edge of the screen.
Before I will do this not very nice procedure again to fix the leakage (the white foil or the backlight got a bit misaligned while closing..), I wanted to get a new backlight to replace the old one. Does anyone know where I can buy one for an T23 SXGA+ display from Hitachi?