= Solution
The radial factor
$$
r^n\exp\left(-\frac{eBr^2}{4\hbar}\right)
$$
is sharply peaked, for large $n$, near
$$
r_n^2=\frac{2\hbar n}{eB}.
$$
Requiring the orbit to lie inside a disk of radius $R$ permits
$$
0\leq n\lesssim\frac{eBR^2}{2\hbar}.
$$
Thus the total degeneracy at fixed $k_z$ is, up to the negligible boundary correction,
$$
\boxed{D=\frac{eBR^2}{2\hbar}}.
$$
Since the disk has area $\pi R^2$, the <degeneracy of a Landau level> per unit area is
$$
\boxed{\frac D{\pi R^2}=\frac{eB}{2\pi\hbar}}.
$$
For either sign of the charge-field product, replace $eB$ by $|eB|$.
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