Thank you both for the quick replies!
MAX7221 would be a better chip for this situation because it's supposed to have lower EMI.
What makes you say that?
The description on the data sheet. It's a guess, but for a few bucks I'm tempted to try it.
Any high speed digital signals generate signals that can be picked up on audio circuits, it is quite common.
You need to make sure your analogue ground and your digital ground are kept separate and only joined at one point. Keep digital lines as far away as possible from analogue lines. Make sure the digital power supply is separate or at least decoupled from the analogue supply.
You can also limit the bandwidth of the audio circuits so they don't pick up as much.
Yes tiding up that rats nest will help but without correct layout it will not be enough.
The analog and digital grounds are separate and joined at one point near the volume control IC. Both DGND and AGND on that IC are connected to the analog ground as per (http://www.analog.com/library/analogDialogue/Anniversary/12.html
). The system as a whole is a "star of stars". I've used Dave Davenport's Audio Component Grounding and Interconnection guide to get me this far with the grounds.
The amp has its own 12V supply and I have a +/- 5V supply for 5V to digital and +/- 5V to analog, so the +5V line is being shared between the digital and analog ICs. It's decoupled when it hits the board with a 10uF and .1uF in parallel to ground and each IC is decoupled with a .1uF near the supply pin to ground as well. I read the page on your site on decoupling and it was helpful.
Does it help to say that the amplitude of the noise increases as the number of illuminated LEDs increases?
*edit: added photo of ground layout