In this talk I show that given a compact Riemannian $3$-manifold with boundary with an $L^2$-bound on its Ricci tensor, an $L^4$-bound on the second fundamental form of the boundary, and a volume radius bound away from zero, the manifold can be covered by local coordinate patches (of controlled size) in which the metric components $g_{ij}$ are $H^2$-regular. The proof follows by extending the theory of Cheeger-Gromov convergence to include manifolds with boundary in the above low regularity setting. The main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate, and a rigidity argument using manifold doubling.