Let T be a compact torus and (M,\omega) a
Hamiltonian T-space. In a previous paper, the authors showed that the
T-equivariant K-theory of the manifold M surjects onto the ordinary integral
K-theory of the symplectic quotient M \mod T of M by T, under certain technical
conditions on the moment map. In this paper, we use equivariant Morse theory to
give a method for computing the K-theory of the symplectic quotient by
obtaining an explicit description of the kernel of the surjection \kappa:
K^*_T(M) \onto K^*(M \mod T). Our results are K-theoretic analogues of the work
of Tolman and Weitsman for Borel equivariant cohomology. Further, we prove that
under suitable technical conditions on the T-orbit stratification of M, there
is an explicit Goresky-Kottwitz-MacPherson (``GKM'') type combinatorial
description of the K-theory of a Hamiltonian T-space in terms of fixed point
data. Finally, we illustrate our methods by computing the ordinary K-theory of
compact symplectic toric manifolds, which arise as symplectic quotients of an
affine space \C^N by a linear torus action.