Magnetic penetration depth in KFe$_2$As$_2$

KFe$_2$As$_2$ is the end member of the (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ series of iron-pnictide superconductors. Previous measurements have suggested that the gap structure of KFe$_2$As$_2$ is nodal, either accidentally or due to the symmetry of the pairing state, in contrast to the rest of the series. Our measurements of the magnetic penetration depth, $\lambda(T)$ of (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ for $x=0.85,0.91,1.0$ all show $\lambda(T)$ saturates at low temperature, which is indicative of either a small energy gap or a small amount of residual disorder. To differentiate between these possibilities we studied both the effect of non-linear, field dependent, corrections, as well as the effect of disorder, on $\lambda(T)$.