With the strong connotation of the verse being doggerel. The person so afflicted is, of course, a metromaniac. In honesty, I should mark that definition with (arch.) if not (obs.), but I have hope we can revive it still. Certainly, I know enough metromaniacs that it seems still useful. At the root is ancient Greek metron, measure/meter, as in poetic meter -- which is, after all, measuring the line.
One glimpse of her set off his metromania, and he ran home to write a dozen sonnets to her eyes.