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Spherical Jeans modeling is widely used to estimate mass profiles of systems from star clusters to galactic stellar haloes to clusters of galaxies. It derives the cumulative mass profile, M(<r), from kinematics of tracers of the potential under the assumptions of spherical symmetry and dynamical equilibrium. We consider the application of Jeans modeling to mapping the dark matter distribution in the outer reaches of the Milky Way using field halo stars. We present a novel non-parametric routine for solving the spherical Jeans equation by fitting B-splines to the velocity and density profiles of halo stars. While most implementations assume parametric forms for these profiles, B-splines provide non-parametric fitting curves with analytical derivatives. Our routine recovers the mass profiles of equilibrium systems with flattened haloes or a stellar disc and bulge excellently (<~ 10% error at most radii). Tests with non-equilibrium, Milky Way-like galaxies from the Latte suite of FIRE-2 simulations perform quite well (<~ 15% error for r <~ 100 kpc). We also create observationally motivated datasets for the Latte suite by imposing selection functions and errors on phase space coordinates characteristic of Gaia and the DESI Milky Way Survey. The resulting imprecise and incomplete data require us to introduce an MCMC-based subroutine to obtain deconvolved density and velocity dispersion profiles from the tracer population. With these observational effects taken into account, the accuracy of the Jeans mass estimate remains at the level 20% or better.