The machine is working as it should, only that the drive led is constantly active. And I have absolutely no idea on to what I should look for. Any suggestions?
The floppy drive LED is driven from the MTR0 signal, the same signal that tells the drive motor to switch on. Meaning that while the line is low to activate the motor on the internal drive, the LED turns on as well.
So the question is that when the floppy drive LED is on, then is the internal drive motor on as well? It's very quiet and you may have to put your ear directly on the A600 to hear it.
If the drive motor is turning on and off as it should but the LED remains on, then look at the LED driver circuit. Which is simply a PNP transistor, Q632. The circuit is configured so when MTR0 is low, drawing current from Q632 base, it causes the transistor to conduct 5V to drive the LED.
If the transistor has been damaged, it'll usually fail short circuit, which could explain the LED being permanently on.
In which case the solution is to replace Q632. The original device is a 2N3906 or MMBT3906. I don't have one in front of me, but it's probably a SOT-23 package. i.e. the slightly larger surface mount type, not the smaller SOT-323. Any other general purpose PNP transistor in the same package will be fine as well, e.g. BC857. They're easy to source and cost a few cents each.