I've assembled a bunch of Avondale-based amplifiers now, and I've found the issue of powering the switch's LED to be the most consistently challenging task. That seems counterintuitive to me, but it is what it is.
First step is to recognize that the switch and the LED are separate entities. They occupy the same space, but will be in two completely different circuits.
The switch controls the power coming into your amp, and it doesn't care if the LED is hooked up at all. It's not polar at all. The "hot" side of your power input should go to the fuse, then to one side of the switch. Other other side of the switch goes to your amplifier's hot input. (There's a good chance that your amp board doesn't care which is hot, in which case, just hook it up to one side.) The amp's second mains connection comes directly from your mains neutral.
Then comes the LED, which can be a bit tricky. It needs to get power from
somewhere, and there are numerous options. Note that it wants DC, so you cannot hook it up directly to your mains. Also, it needs a resistor to reduce the current, so that you don't blow the LED.
Given the low current requirements and stable load, you may be able to connect your LED to your pre-amp's power supply (after rectification and smoothing). I just took a very quick look at it myself, and I didn't see anything obvious. It's worth you checking closer, and maybe asking over on the diyaudio forum.
Alternatively, you could create a simple power supply for the LED, using a rectifier diode and smoothing capacitor. Google is your friend! You also need to experiment with resistor values until you find one that displays the desired brightness. Start with something really big, like 200 k ohms (which will likely prevent the LED from lighting). Try progressively smaller values (perhaps dropping by 1/3 each time), until the LED is a reasonable brightness. If you go this route, then this little LED power supply will also be connected between the output of the switch and your mains neutral.
And just to be clear (as I'm unsure of your knowledge base), the mains neutral is not the same thing as the mains ground.
If I can find a simple PS schematic for the LED, I'll post it later.