  The charging of a quantum box, coupled to a lead by tunneling through a
single resonant level, is studied near the degeneracy points of the Coulomb
blockade. Combining Wilson's numerical renormalization-group method with
perturbative scaling approaches, the corresponding low-energy Hamiltonian is
solved for arbitrary temperatures, gate voltages, tunneling rates, and energies
of the impurity level. Similar to the case of a weak tunnel barrier, the shape
of the charge step is governed at low temperatures by the non-Fermi-liquid
fixed point of the two-channel Kondo effect. However, the associated Kondo
temperature TK is strongly modified. Most notably, TK is proportional to the
width of the level if the transmission through the impurity is close to unity
at the Fermi energy, and is no longer exponentially small in one over the
tunneling matrix element. Focusing on a particle-hole symmetric level, the
two-channel Kondo effect is found to be robust against the inclusion of an
on-site repulsion on the level. For a large on-site repulsion and a large
asymmetry in the tunneling rates to box and to the lead, there is a sequence of
Kondo effects: first the local magnetic moment that forms on the level
undergoes single-channel screening, followed by two-channel overscreening of
the charge fluctuations inside the box.
