There's no reliable way to do this,
screen.height will always return the height of the primary monitor, doesn't matter how many monitors you have, so that's not going to help at all.
Using screenX/Y doesn't really help either, as there are no indications of when one screen ends and another begins, and trying to calculate it would be hard as there's no guarantee the user starts out with the browser at the top left in one of the screens, or that the browser takes up the entire screen, and you can't get the dimensions for any other monitor than the primary screen, so there's no way to know the size of the other screens.
In other words, it's going to be hard to do this, if not impossible, and any solution will probably not be very reliable or portable, so you're better of just sticking with the browser window and detecting changes to that, which is easy and generally all you need.
window.screen.height. Now, if I have dual monitors, and the two monitors are different sizes, and I move the browser window from one monitor to the other, is there any way to detect that, beyond polling the window size at intervals in time?