The interaction of atomic clusters with short, intense pulses of laser light to form extremely hot, dense plasmas has attracted extensive experimental and theoretical interest. The high density of atoms within the cluster greatly enhances the atom-laser interaction, while the finite size of the cluster prevents energy from escaping the interaction region. Recent technological advances have allowed experiments to probe the laser-cluster interaction at very high photon energies, with interactions much stronger than suggested by theories for lower photon energies. We present a model of the laser-cluster interaction that uses nonperturbative $R$-matrix techniques to calculate inverse bremsstrahlung and photoionization cross sections for Herman-Skillman atomic potentials. We describe the evolution of the cluster under the influence of the processes of inverse bremsstrahlung heating, photoionization, collisional ionization and recombination, and expansion of the cluster. We compare total energy absorbed with the Hamburg experiment of Wabnitz et al. [Nature 420, 482 (2002)] and ejected electron spectra with Laarmann et al. [Phys. Rev. Lett. 95, 063402 (2005)].