My laptop screen is broken and am using an external monitor.
However, today the monitor switched itself off, and the amber standby light is on. I have pressed FN and F4 to reawaken the external monitor (which worked before), but at the moment it's not working and the monitor remains on standby
It's difficult to tell, but the amber standby light means pretty much what its name suggests, i.e. that the monitor has gone into standby mode because it hasn't received any input for a certain amount of time.
To quote The I.T. Crowd, have you tried turning if off and on again?
If you turn it off, unplug it from the laptop and then turn it back on again, does it say something like NO SIGNAL?