  We prove an integral inequality, first proven by Bray-Kazaras-Khuri-Stern on
three-dimensional asymptotically Euclidean (AE) manifolds, for the ADM mass on
AE K\"ahler manifolds $(X,g,J)$ of all complex dimensions. This inequality
bounds the ADM mass from below by an integral of the scalar curvature and the
Hessian of any holomorphic coordinate function on $X$ arising in Hein-LeBrun's
work on the ADM mass on K\"ahler manifolds. Using this, we transfer the proof
of the stability of the Positive Mass Theorem from Dong-Song and
Kazaras-Khuri-Lee from three-dimensional manifolds to AE K\"ahler manifolds. We
conclude that, for any such sequence of AE K\"ahler manifolds, we can find
subsets of each with vanishing boundaries in the limit such that the
complements converge to Euclidean space in the pointed Gromov-Hausdorff sense.
If we impose a uniform lower bound on the Ricci curvature, the result holds
without taking the complement of a sequence of vanishing sets. Finally, we find
three new families of AE K\"ahler manifolds with vanishing mass in the limit
for which the stability results apply.
