In the $n+mK$ dimensional differentiable manifold $E^*$ ($K$-Hamilton space) special coordinate transformations are allowed. In $T^*(E^*)\otimes T^*(E^*)$ the metric tensor is given, and using the nonlinear connection $N,T(E^*)$ may be decomposed in $K+1$ orthogonal subspaces (with respect to $G$): $T_h(E^*)$ and $_{(\alpha)}T_V(E^*)$, $\alpha=\overline{1,K}$. In $T(E^*)$ a strongly distinguished connection is introduced in such a way that $Y$ and $\nabla_XY$ belong to the same subspace of $T(E^*)$, $\forall X,Y\in T(E^*)$. The law of transformation of connection coefficients is given. For the metrical and recurrent case the connection coefficients and the torsion tensor are determined.