Previously, we have performed the spin-polarized hybrid-density functional theory (HUDFT) calculations for elucidating magnetic properties of the two-dimensional (2D) K2NiF4 and K2CuF4 solids. In Part I, it has been concluded that the half-and-half-(HH-) type HUDFT method is one of the best calculation methods for these species. On the other hand, in Part II, we have demonstrated that potential curves for cluster models of K2CuF4 and KCuF3 are reasonably calculated by the HH-type HUDFT method under the approximate spin projection, and the lattice distortion resulted by Jahn-Teller effect is expressed as the second-order polynomial. In this study, we pay attention to the three-dimensional (3D) magnetic interactions in KNiF3. Our effective exchange integral J(ab) schemes for 3D cluster models such as KNi8F12 (4) with and without point charges have provided the reasonable J(ab) values (-30.24-34.48 cm-1), in comparison with the experimental one (-30.58 cm-1). The 3D magnetic interactions have been investigated from viewpoints of the Mulliken spin density and charge density populations, the natural orbital analysis, and chemical indices. Point charges located in positions of fluorine anion adjacent to cluster model have reproduced J(ab) values well. Roles of potassium as counter cation in KNiF3 solid were also investigated. It was concluded that potassium has a role of stabilizing the 3D magnetic structures. Finally, the mutual relationships between broken-symmetry and symmetry-adapted approaches are discussed on the basis of chemical indices. (C) 2004 American Institute of Physics.