  We investigate the small but finite amplitude solitary Kinetic Alfv\'{e}n
waves (KAWs) in low $\beta$ plasmas with superthermal electrons modeled by a
kappa-type distribution. A nonlinear Korteweg-de Vries (KdV) equation
describing the evolution of KAWs is derived by using the standard reductive
perturbation method. Examining the dependence of the nonlinear and dispersion
coefficients of the KdV equation on the superthermal parameter $\kappa$, plasma
$\beta$ and obliqueness of propagation, we show that these parameters may
change substantially the shape and size of solitary KAW pulses. Only
sub-Alfv\'enic, compressive solitons are supported. We then extend the study to
examine kinetic Alfv\'en rogue waves by deriving a nonlinear Schr\"{o}dinger
equation from {the KdV} equation. Rational solutions that form rogue wave
envelopes are obtained. We examine how the behavior of rogue waves depends on
the plasma parameters in question, finding that the rogue envelopes are lowered
with increasing electron superthermality whereas the opposite is true when the
plasma $\beta$ increases. The findings of this study may find applications to
low $\beta$ plasmas in astrophysical environments where particles are
superthermally distributed.
