Does the screen change at all when you power it on? Sometimes it can be hard to tell, but if you're getting a backlit black screen, then the display itself is probably still working fine. It's pretty rare for the GPU to fail on the 2013 27-inch iMacs, so I'd be surprised if that was the case. Sometimes a failing internal hard drive can cause the symptoms you're describing, though, especially a failing Fusion Drive. Unfortunately I think the only way forward at this point would be to open up the device and test the internal components. Good luck!