Thanks for sharing your experience Alexmam. This is a very enigmatic issue for sure. I basically cannot handle any LED monitor either. Anyone in the office that has one, I almost always convince them to come to my desk instead. If I don't, I will have an ocular-like migraine for pretty much the rest of the day. The pain is almost immediate when I look at their monitor too.
I visited an neuro-ophthalmologist this month, and in dilating my pupils and looking in, he said my retina and head of the optic nerve looks OK (that's good at least! Good to know these painful monitors aren't causing any eye damage). He suspects the pain is actually coming from the brain (like the visual processing center). Some of his associates have said that some people have gained some relief from the FL-41 tint. I think a company called Theraspecs in the US makes them.
It's also been suggested that I may have a form of Irlen Syndrome (Scotopic Sensitivity Syndrome). I'm a bit skeptical, but it at least makes sense in theory. For now, I'm just sticking with my U2410 LCD monitor on 0% brightness with f.lux enabled. That seems to be the most tolerable thing I have found so far.
I can use an iPhone 6S+ fine for a few minutes each day as well. That doesn't seem to bother me as it does some others on here.