We study infinite-horizon Distributionally Robust (DR) control of linear dynamical systems with quadratic cost. The probability distribution of the disturbances is unknown and possibly time-correlated, residing within a Wasserstein-2 ambiguity set centered around a nominal distribution. We aim to identify a control policy that minimizes the worst-case expected regret, representing the excess cost incurred by a causal control policy compared to a non-causal counterpart with access to future disturbances. While the optimal policy lacks a finite-dimensional state-space realization, we show that it can be characterized through a finite-dimensional parameter and develop an exponentially-convergent fixed-point algorithm that finds the optimal controller in the frequency domain. We present an efficient algorithm that finds the best controller for any given state-space dimension, thereby circumventing the computational challenges associated with finite-horizon problems that require solving an SDP whose dimension scales with the time horizon.