We present a measurement of the atmospheric νe spectrum at energies between 0.1 and 100 TeV using data from the first year of the complete IceCube detector. Atmospheric νe originate mainly from the decays of kaons produced in cosmic-ray air showers. This analysis selects 1078 fully contained events in 332 days of live time, and then identifies those consistent with particle showers. A likelihood analysis with improved event selection extends our previous measurement of the conventional νe fluxes to higher energies. The data constrain the conventional νe flux to be 1.3+0.4−0.3 times a baseline prediction from a Honda’s calculation, including the knee of the cosmic-ray spectrum. A fit to the kaon contribution (ξ) to the neutrino flux finds a kaon component that is ξ=1.3+0.5−0.4 times the baseline value. The fitted/measured prompt neutrino flux from charmed hadron decays strongly depends on the assumed astrophysical flux and shape. If the astrophysical component follows a power law, the result for the prompt flux is 0.0+3.0−0.0 times a calculated flux based on the work by Enberg, Reno, and Sarcevic.