LED's emitting high pitched noise
It's impossible for an LED*  to emit noise (more than once). 

This is a problem with the power supply or driver circuit.  

If these are 120V screw-in LEDs, then the built-in driver is the problem and replacing the whole unit is hte only option.

If this is a "plug-n-play" LED conversion of a fluorescent tube, then the old fluorescent ballast is still in play, and that is surely the source of the noise.  It should be replaced with a ballast-bypass aka direct-wire LED, which will remove the ballast from the circuit altogether.  



* If you think I mean the consumer products sold at Ikea and Walmart that replace incandescent bulbs, no. I make that clear later in my answer.  I mean LED components. Which is very relevant to trophy cabinets, because they are typically LED "strips" comprised of actual LEDs, and separate power supplies.