We have modeled the injection and acceleration of pickup ions at the solar wind termination shock and investigated the parameters needed to produce the observed anomalous cosmic-ray (ACR) fluxes. A nonlinear Monte Carlo technique was employed that in effect solves the Boltzmann equation and is not restricted to near-isotropic particle distribution functions. This technique models the injection of thermal and pickup ions, the acceleration of these ions, and the determination of the shock structure under the influence of the accelerated ions. The essential effects of injection are treated in a mostly self-consistent manner, including effects from shock obliquity, cross-field diffusion, and pitch-angle scattering. Using recent determinations of pickup ion densities, we are able to match the absolute flux of hydrogen in the ACRs by assuming that pickup ion scattering mean free paths, at the termination shock, are much less than an AU and that modestly strong cross-field diffusion occurs. Simultaneously, we match the flux ratios He^+/H^+ or O^+/H^+ to within a factor ~5. If the conditions of strong scattering apply, no pre-termination-shock injection phase is required and the injection and acceleration of pickup ions at the termination shock are totally analogous to the injection and acceleration of ions at highly oblique interplanetary shocks recently observed by the Ulysses spacecraft. The fact that ACR fluxes can be modeled with standard shock assumptions suggests that the much discussed ``injection problem'' for highly oblique shocks stems from incomplete (either mathematical or computer) modeling of these shocks rather than from any actual difficulty shocks may have in injecting and accelerating thermal or quasi-thermal particles.