This paper solves the inverse dynamics of a tethered kite analytically. Specifically, the paper presents a procedure for determining the angle of attack, induced roll angle, and tether tension magnitude needed to achieve a desired combination of translational kite position, velocity, and acceleration. The focus of the paper is on energy harvesting kites. However, the underlying approach is applicable to other kite systems, such as kites for propulsion (e.g., SkySails). Solving inverse kite dynamics analytically is valuable for trajectory optimization, online state estimation, and the analysis of fundamental limitations on kite maneuvers. Previous work in the literature presents several models of kite dynamics, with varying degrees of fidelity and complexity. However, the nonlinearity of these models often makes them difficult to use for optimization, estimation, and control. The paper shows that, under reasonable assumptions, inverse kite dynamics can be solved in terms of the roots of a fourth-order polynomial function of angle of attack. This function has a geometric interpretation, providing insight into the multiplicity of resulting solutions. Moreover, for special cases including a kite with non-cambered wings, these solutions can be computed analytically. A simulation validates the success of the proposed approach in computing inverse kite dynamics for a cross-current trajectory.