= Solution
Use logarithms to base two, so all <Von Neumann entropies> and <Shannon entropies> are measured in bits. Write the nonzero <eigenvalues> of $\rho_i$ as $\lambda_{ij}$. Orthogonality of the supports makes the mixture a <block diagonal matrix>, with eigenvalues $p_i\lambda_{ij}$. Therefore
$$
\begin{aligned}
S(\rho)&=-\sum_{i,j}p_i\lambda_{ij}\log_2(p_i\lambda_{ij})\\
&=-\sum_i p_i\log_2p_i\sum_j\lambda_{ij}
-\sum_i p_i\sum_j\lambda_{ij}\log_2\lambda_{ij}\\
&=\boxed{H(p)+\sum_i p_iS(\rho_i)}.
\end{aligned}
$$
Each $\rho_i$ has <trace> one. Terms with zero probability or zero eigenvalue vanish under the convention $0\log0=0$. This proves the <entropy of an orthogonal quantum mixture>.
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