It doesn’t surprise me at all that the LED screens do you in. Note that this is NOT a function of LED light–LED light is indeed flicker-free. The issue is that they don’t know how to make the LED lights work at different brightness levels, or at least not inexpensively. So they way they control brightness on an LED screen is to duty cycle the LED light on and off rapidly. The longer the period of darkness, the less bright the light seems. They flash the light at a rate (120-180 times a second) that no one can physically see the flickering. However, there has been some discussion that people with migranes have a much higher “flicker fusion threshold”, i.e. their brains are bothered by flickering at very fast rates even though they can’t physically see the flashing occurring.
My advice to anyone having problems with an LED monitor is to turn the brightness to 100% and then use a video card control panel to bring it back down and see if that helps at all. This won’t work on every monitor, as some still flicker even at 100% brightness(my LG IPS monitor does not). This would probably screw up color saturations, so if your and artist who needs color purity I’d go with an IPS LCD like the Dell U2410 (or better).
The reason IPS is better is that it can natively display all the colors it needs. The cheaper TN panel monitors (most in existance) use dithering to display some colors. Meaning they rapidly flash two colors on top of each other and your brain then makes it out as a 3rd color. Again, this is happening so quickly that it is impossible to see, but the theory is that the migraine brain can be bothered anyway.