We calculate the amount of entanglement for bound and doubly excited singlet states below the second, third and fourth ionization thresholds in planar helium. Furthermore, we investigate the dynamics of entanglement in nondispersive two-electron wave packets generated by the resonant coupling of frozen planet states of helium. We show that it is possible to obtain a two-electron nondispersive wave packet for which the time-average entanglement is greater than for each of the unperturbed coupled states.