  Narrow Fe K$\alpha$ fluorescent emission lines arising at $\sim$kpc-scale
separations from the nucleus have only been detected in a few AGN. The
detections require that the extended line emission be spatially resolved and
sufficiently bright. Compared to narrow Fe K$\alpha$ lines arising closer to
the nucleus, they have much lower fluxes but show substantially larger
equivalent widths, EW$_{\rm Fe K\alpha}$. We show that, in the optically-thin
limit, a purely analytical argument naturally predicts large, EW$_{\rm
FeK\alpha}\sim$1 keV, values for such lines, regardless of the details of
equivalent hydrogen column density, $N_H$, or reprocessor geometry. Monte Carlo
simulations corroborate this result and show that the simple analytic EW$_{\rm
FeK\alpha}$ prescription holds up to higher $N_H$ approaching the Compton-thick
regime. We compare to $Chandra$ observations from the literature and discuss
that our results are consistent with the large EW$_{\rm FeK\alpha}$ values
reported for local AGN, for which the line is detected in extended, up to
$\sim$kpc-scale, regions. We argue that large EW$_{\rm FeK\alpha}$ from
kpc-scale regions in AGN should be ubiquitous, because they do not depend on
the absolute luminosity of the central X-ray source, and are measured only
against the scattered continuum. We predict values to be of the order of
$\sim$1 keV or larger, even for covering factors $\ll$1, and for arbitrarily
small column densities. We propose that the large-scale molecular material that
is now routinely being detected with the Atacama Large Millimeter/Submillimeter
Array (ALMA) may act as an extended X-ray scattering reprocessor giving rise to
$\sim$kpc-scale Fe K$\alpha$ emission.
