Two-electron wave packets found in driven planar helium follow a classical periodic orbit without spreading and are localized in nonlinear resonance islands of the classical phase space. The general mechanism that produces these nondispersive wave packets is the near-resonant coupling between highly correlated, asymmetric, and doubly excited states of the atom with an external periodic electromagnetic field. We provide a full characterization of these two-electron nondispersive wave packets found in the Floquet spectrum of driven planar helium. This includes the characteristic energy spectrum, relatively long lifetimes, and the identification of the resonance states of helium that are responsible for their formation. This is achieved with the help of an efficient quantum treatment of driven planar helium which combines Floquet theory, complex rotation, and the representation of the Hamiltonian in a set of four coupled harmonic oscillators.