I'm pretty sure that it's the backlight that's busted because, under the right lighting conditions, I can still see the stuff displayed on the screen, but it's very faint.
I've tried to manually control the screen power saving mode by blindly typing
vbetool dpms onat the console, but this has no visible effect.
The box is otherwise just fine, and it still functions as a backup server, web server, firewall, printer server etc. It's just that I can't use it as a desktop.
What a bummer.
I'm still considering my options. I've been thinking of getting a new laptop for my wife for quite a while, since the (extended) warranty on her current, 4 year old, laptop expired, and she does need a lighter machine (2.7kg is pretty heavy...). But a new laptop would cost more than fixing my box.
To be continued.