We present calculations of excitation energies within the time-dependent density functional theory (TDDFT) extension of frozen-density embedding (FDE) using reconstructed accurate embedding potentials. Previous applications of FDE showed significant deviations from supermolecular calculations; our current approach eliminates one potential error source and yields excitation energies of generally much better agreement with Kohn-Sham-TDDFT. Our results demonstrate that the embedding potentials represent the main error source in FDE-TDDFT calculations using standard approximate kinetic-energy functionals for excitations localized within one subsystem.