This paper presents the formulation and results for a control-aware optimization of the combined geometric and structural design of an energy-harvesting underwater kite. Because kite-based energy-harvesting systems, both airborne and underwater, possess strong coupling between closed-loop flight control, geometric design, and structural design, consideration of all three facets of the design within a single co-design framework is highly desirable. However, while prior literature has addressed one or two attributes of the design at a time, the present work constitutes the first comprehensive effort aimed at addressing all three. In particular, focusing on the goals of power maximization and mass minimization, we present a co-design formulation that fuses a geometric optimization tool, structural optimization tool, and closed-loop flight efficiency map. The resulting integrated co-design tool is used to address two mathematical optimization formulations that exhibit subtle differences: a Pareto optimal formulation and a dual-objective formulation that focuses on a weighted power-to-mass ratio as the techno-economic metric of merit. Based on the resulting geometric and structural designs, using a medium-fidelity closed-loop simulation tool, the proposed formulation is shown to achieve more than three times the power-to-mass ratio of a previously published, un-optimized benchmark design.