  We present a measurement of the atmospheric $\nu_e$ spectrum at energies
between 0.1 TeV and 100 TeV using data from the first year of the complete
IceCube detector. Atmospheric $\nu_e$ 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 $\nu_e$ fluxes to higher energies. The
data constrain the conventional $\nu_e$ 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 ($\xi$) 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.
