= Solution
Work in units with $\hbar=1$. For unit mass and frequency, the <two-dimensional isotropic harmonic oscillator> has
$$
H=\frac12(p_x^2+x^2)+\frac12(p_y^2+y^2)
=a_x^\dagger a_x+a_y^\dagger a_y+1.
$$
The product number states are therefore energy eigenvectors:
$$
|n_x,n_y\rangle,
\qquad
E_{n_x,n_y}=n_x+n_y+1.
$$
For the level $E=N+1$, the possibilities are
$$
(n_x,n_y)=(N,0),(N-1,1),\ldots,(0,N),
$$
so its degeneracy is
$$
\boxed{N+1=E}.
$$
Restoring units multiplies every energy by $\hbar\omega$.
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