I did NOT move the logic board. The original hard disk could be taken out with a little(!) force. The new replacement SSD was much thinner and was not problem to slide in.
We changed only the three 25V 220uF capacitors (two of them looked broken), and the backlight was working again. Thanks for the description! Spent 0,24€ and around an hour for the fix. As a beginner in soldering:-)