= Solution
Interpret the stated <Fermi energy> as measured above the bottom $E_0$ of the occupied subband, and take a degenerate, nearly zero-temperature gas. Then
$$
n_s=\int_{E_0}^{E_F}D_{2D}(E)\,dE
=\frac{m^*}{\pi\hbar^2}(E_F-E_0).
$$
For a $0.100\,\mathrm{eV}$ filling energy,
$$
\boxed{n_s=(2.80\times10^{13})(0.100)
=2.80\times10^{12}\,\mathrm{cm^{-2}}
=2.80\times10^{16}\,\mathrm{m^{-2}}.}
$$
A Fermi energy referenced instead to an unspecified bulk conduction-band edge would not determine this number without the subband offset. The assumption that only one subband is occupied is imposed by the question; at this comparatively large density, an actual confinement potential should be checked before applying it to a device.
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