It appears the logic board will need repairs or just replacing it (logic board P/N 661-5544). Given the age of the display, it’s likely a weak cap or cold solder joint is the issue.
See if you can find a set of the '''''LED Cinema Display (27-inch) Schematics & Boardview''''' to help you isolate the area of theologic board that needs service.
Using the Mac you can now check the displays firmware is upto date: [https://support.apple.com/kb/DL1613?viewlocale=en_US&locale=en_US|LED Cinema Display Software Update 1.1] This addresses a speaker issue it had.