I've never seen a power cord go bad like that, but since it's the easiest thing to try, you may as well try another one. Another thing to check would be if the display appears on the iMac when plugged in. Last time I looked at one of these it was rather hard to tell that there was a display when using a flashlight on it until I started moving the mouse around. I realized that the Mac was detecting the display and outputting an image but the backlight wasn't working. This particular monitor would slightly flicker when plugging it into the Mac.
If you're absolutely certain there's no powering on, then it may be the power supply. [https://dreamsupport.us/downloads/apple-mac/Apple%20Service%20Manuals/Service%20Manuals/Displays/LED%20Cinema%20Display%20(27-inch)/led_cinema_display_27.pdf|Here's a link] to the Apple Technician Guide for this model.