A toric hyperkähler manifold is defined as a hyperkähler quotient of the flat quaternionic space HN by a subtorus of the real torus TN. The purposes of this paper are to construct compact complex submanifolds of toric hyperkähler manifolds, and to show that our hyperkähler manifold is a resolution of singularities of an affine algebro-geometric quotient. We also show that these submanifolds are biholomorphic to Delzant spaces, which are Kähler quotients of CN by subtori of TN. Finally, we apply these results to determining whether complex structures on our hyperkähler manifold are equivalent.