Hi,
Same thing happened on my E39 after jumping it while the monitor was turned on. The display went blank but all radio functions worked.
Some Google-Fu pointed to a blown transistor on the boardmonitor PCB, I pulled the screen apart and sure enough, 2 transistors were fried. iirc they were part of the 5V power-on.
Anyway I ordered new transistors ($10 from mouser) & tried to resolder them.... but I'm a novice at soldering and just made a hash of the whole thing.
In the end I bought a 2nd hand screen off TradeMe for $100, now I have a working display.
Unfortunately the replacement was out of an E38 so the viewing angle isn't quite right...looks a little washed out at night. Can be adjusted via pots though quite painful process from what I have read. :).
The transistor # is C3518 -K57, I have a couple spare if you need any...they came in pack of 5. I can also supply a photo of the old board just ask.