An analytical Bessel-sine series solution is presented for inviscid, incompressible, axisymmetric, annular impinging jets. The inflow velocity profile is relatable to an aircraft engine thrust reverser, and has a velocity deficit as well as nonzero vorticity function at the inner radius. As a consequence, inviscid recirculation appears at the impingement corner, the strength of which is made determinate by assuming a local patch of constant vorticity function. Results are first compared against prior round jet solutions, with rigorous convergence and accuracy assessment, before extension to annular impinging jets. Finally, while the Bessel-sine series are consistent with the governing equations and boundary conditions, computational issues arise with steep near wall gradients of a composite parameter that is inherent to the methodology, necessitating extremely fine mesh resolution. A hybrid of finite differences and Bessel series is proposed to address these numerical issues.