We present a measurement of the atmospheric $ u_e$ spectrum at energies between 0.1 TeV and 100 TeV using data from the first year of the complete IceCube detector. Atmospheric $ u_e$ originate mainly from the decays of kaons produced in cosmic-ray air showers. This analysis selects 1078 fully contained events in 332 days of livetime, then identifies those consistent with particle showers. A likelihood analysis with improved event selection extends our previous measurement of the conventional $ u_e$ fluxes to higher energies. The data constrain the conventional $ u_e$ flux to be $1.3^{+0.4}_{-0.3}$ times a baseline prediction from a Hondas calculation, including the knee of the cosmic-ray spectrum. A fit to the kaon contribution ($xi$) to the neutrino flux finds a kaon component that is $xi =1.3^{+0.5}_{-0.4}$ 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.
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