This paper presents a method for determining the six-degree-of-freedom (DOF) transformation between a camera and a base frame of interest, while concurrently estimating the 3D base-frame coordinates of unknown point features in the scene. The camera observes the reflections of fiducial points, whose base-frame coordinates are known, and reconstruction points, whose base-frame coordinates are unknown. In this paper, we examine the case in which, due to visibility constraints, none of the points are directly viewed by the camera, but instead are seen via reflection in multiple planar mirrors. Exploiting these measurements, we analytically compute the camera-to-base transformation and the 3D base-frame coordinates of the unknown reconstruction points, without a priori knowledge of the mirror sizes, motions, or placements with respect to the camera. Subsequently, we refine the analytical solution using a maximum-likelihood estimator (MLE), to obtain high-accuracy estimates of the camera-to-base transformation, the mirror configurations for each image, and the 3D coordinates of the reconstruction points in the base frame. We validate the accuracy and correctness of our method with simulations and real-world experiments.