  We study left-invariant Killing $k$-forms on simply connected $2$-step
nilpotent Lie groups endowed with a left-invariant Riemannian metric. For
$k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing
forms on the factors of the de Rham decomposition. Moreover, on each
irreducible factor, non-zero Killing $2$-forms define (after some modification)
a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms
arise only if the Riemannian Lie group is naturally reductive when viewed as a
homogeneous space under the action of its isometry group. In both cases, $k=2$
or $k=3$, we show that the space of Killing $k$-forms of an irreducible
Riemannian 2-step nilpotent Lie group is at most one-dimensional.
