Since $E_G \gg k_B T$, we know that $E_C-E_D < E_G$ and $E_A - E_V < E_G$ and so we are in the regime of full dopant ionization. Therefore, $n_D = n_A = 0$.
Since $E_G \gg k_B T$, we can only use the law of mass action.
But the question offers us another piece of information - we are around $|N_D-N_A| ~ n_i$.
That means that we are near the transition between extrinsic and intrinsic regimes.
However, we also know that dopants energies are quite small such that $E_C-E_A \ll E_G$ and $E_A-E_V \ll E_G$.
That means that we expect $n_i \ll n_D$ and $n_i \ll n_A$ (since both $n_i$ and dopant ionazition depends exponentially of the corresponding energy differences).
Therefore, we can confidently say that $N_D \gg n_D$ and $N_A \gg n_A$ so we esentially recover the dopant ionization condition.
Thus, neglecting $n_D$ and $n_A$ such that they are both 0, the solutions to the charge balance are:
