= Solution
For an in-plane parabolic band $E(k)=E_0+\hbar^2k^2/(2m^*)$, one state occupies area $(2\pi)^2/A$ in two-dimensional wave-vector space. Including spin degeneracy two and the single GaAs conduction-band valley,
$$
\frac{N(E)}{A}=2\frac{\pi k^2}{(2\pi)^2}=\frac{k^2}{2\pi}
=\frac{m^*}{\pi\hbar^2}(E-E_0).
$$
Differentiation gives the constant two-dimensional <density of states>
$$
\boxed{D_{2D}(E)=\frac{m^*}{\pi\hbar^2}\,\Theta(E-E_0)=\frac{4\pi m^*}{h^2}\,\Theta(E-E_0).}
$$
Using the supplied $h$ and effective mass gives
$$
\boxed{D_{2D}=1.75\times10^{36}\,\mathrm{J^{-1}m^{-2}}
=2.80\times10^{13}\,\mathrm{eV^{-1}cm^{-2}}.}
$$
This counts available one-electron states, not the number actually occupied at arbitrary temperature. Occupation requires the <Fermi-Dirac distribution>; each additional subband would add another step above its own threshold.
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