Where did you read to only ground one end of the shielded cable? That's wrong.
With only one end of your existing cable tied to ground, you have basically built an 'antenna' to pick up spurious noise signals.
When your done, you should have some great ground strap material.
The buzzing is presumably the PWM signal to the LEDs.
You haven't said how the LEDs are connected
but you need to be make sure they don't share a common ground wire with the output signal or power supply. If the Mighty has 3 ground pins, then use one for the power in, one for the LED grounds or LED driver grounds, and one for the audio out.
You also need good decoupling of the power to the LEDs, to decrease the amplitude of the LED switching current in the ground side of the power input.
Other options include:- Increase the PWM frequency you are using for the LEDs to more than 20KHz.
- Add an audio transformer between the line output of the Mighty and the amp. You can get these from the major component distributors. A standard 600 ohm 1:1 transformer should do the job.
What exactly do I do with this ground strap once I have it? Do I run the cable through it, or attach it to something? And how is that braided material better than say, a copper wire to ground?It does a much better job of dissapating 'rf' to ground. You would put it between the DC ground (chassis) and frame ground of your vehicle or water pipe ground in a house. The braided wire is much more flexible as well. 8 MHZ is 'RF', just above the 40 meter band. As the other gentleman suggested, a transformer will help. An inductor (sometimes called a 'choke') between the amp and supply may also help. Use the formula 6.28 * Frequency * L (inductance in Henrys) to calculate XL (Inductive Reactance) in ohms. Of course the higher the XL, acts just like resistor in your circuit. As frequency drops, so to does the XL.