Obnoxious high pitched buzzing sound from LED screen to every DIed instrument on stage.

Sounds like that LED wall is throwing off significant amounts of EMI, likely in violation of applicable regulatory requirements. Assuming you can't change the LED wall, keeping unbalanced runs as short as possible and making sure that cable and equipment shielding is rock solid is going to be your next best option. Note that this likely rules out many of the cheaper DIs, especially cheap active DIs.
 
Way too little information to respond to. Are you using different circuits for screen/video/anything connected to this and stage power/FOH? What kind of screen? perhaps take a photograph and share it here.
What does they always set up directly in front mean? If you turn off the screen sound goes away? turn off the driver for the screen goes away? What is on DI? what works in this situation (guitars not on DI for example?) is it guitars or keyboards? if keyboards on DI have it too, sounds like a power issue. See what I mean? this is a guessing game of where is waldo
 
Strats and other single coil guitars are really great at picking up noise from LED screens. For weight reasons the modular LED screens have loads of plastic and few metal parts, and even computer monitors are sources of noise. Guitars with their pedals to over amplify and distort audio are very prone to this. Why did we screen microphones and use balanced connections and totally ignore hum rejection and interference on guitars?
 
LED video walls are notorius spewers of broadband RF noise. Turn it off and see if the noises go away. If so, the wall needs to be moved, shielded, or you will live with this until you die or change venues.
 
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