It might be worth checking the underneath of the board for dry-joints.
I was wondering why my A1200 floppy drive LED doesnt work and I just noticed that all 4 legs have come loose off the PCB. Once I find the soldering iron I will try to reattach it. The same thing can happen to anything that has been soldered on a PCB especially over time.
Generally if it IS a dry joint you just have to reheat the solder on each leg so that it makes a proper connection. However if its actually torn off the PCB you might have to rewire it to somewhere else on the PCB to repair the problem.
If its the power connector itself thats gone funny you might be able to swap it off one of the other A600s you have, if you are better at soldering than I am. :roll: