  We study the relation between the spin fluctuation and superconductivity in
an heavily hole doped end material KFe$_2$As$_2$. We construct a five orbital
model by approximately unfolding the Brillouin zone of the three dimensional
ten orbital model obtained from first principles calculation. By applying the
random phase approximation, we obtain the spin susceptibility and solve the
linearized Eliashberg equation. The incommensurate spin fluctuation observed
experimentally is understood as originating from interband interactions, where
the multiorbital nature of the band structure results in an electron-hole
asymmetry of the incommensurability in the whole iron-based superconductor
family. As for superconductivity, s-wave and d-wave pairings are found to be in
close competition, where the sign change in the gap function in the former is
driven by the incommensurate spin fluctuations. We raise several possible
explanations for the nodes in the superconducting gap of KFe$_2$As$_2$ observed
experimentally.
