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We study the dynamics of homogeneous and isotropic Friedmann-Lema\^itre-Robertson-Walker cosmological models with positive spatial curvature within the context of mimetic gravity theory by employing dynamical system techniques. Our analysis yields phase-space trajectories that describe physically relevant solutions, capturing various stages of cosmic evolution. We also employ Bayesian statistical analysis to constrain the cosmological parameters of the models, utilizing data from Type Ia supernovae and Hubble parameter data sets. The observational data sets provide support for the viability of mimetic gravity models, which can effectively describe the late-time accelerated expansion of the universe.