We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in [Beskos et al., Stochastic Process. Appl., 2011]. This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in [Beskos et al., Stochastic Process. Appl., 2011], and provides an alternative to a recent proof based on exact coupling techniques given in arXiv:1909.07962. The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. [Stuart, Acta Numer., 2010]. Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, arXiv:1808.01084v3.