I claim that
$$\lVert(a_{i,j})\Vert \le n \Bigl\lVert\sum_i a_{i,i}\Bigr\rVert.$$
Indeed, the map
$$\operatorname{Tr}: M_n(A) \to A: (a_{i,j})\mapsto \sum_i a_{i,i}$$ is easily checked to be completely positive.
Let $\{u_i\}$ be an approximate unit for $A$. Then the matrices $$U_i:= \begin{pmatrix}u_i & 0 & \cdots & 0\\ 0 & u_i & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\cdots & u_i\end{pmatrix} \in M_n(A)$$ determine an approximate unit for $M_n(A)$. By general properties of completely positive maps, $$\lVert\operatorname{Tr}\rVert = \lim_i \lVert\operatorname{Tr}(U_i)\rVert = \lim_i \lVert\nu_i\rVert = n$$ and the desired inequality follows.
