I believe the problem here is the notation. Conway uses the notation
- $H^{(n)}$ to mean the $n$-fold direct sum of $H$, and
- $A^{(n)}$ for the $n$-fold direct sum of the operator $A\in B(H)$.
- $\mathcal{A}$ for the $C^*$ or von Neumann algebra.
(I didn't have Conway's book in my office, but I looked at the preview on Google Books, and searched for "cardinal number," and I found Definition 6.3.(a). I don't know if he also uses the notation $\mathcal{A}^{(n)}$ for the $n$-fold direct sum of $\mathcal{A}$, but I can see how this could be confusing!)
In what you've written, $\mathcal{A}^{(n)}$ should be the $n$-fold amplification of $\mathcal{A}$, i.e., the image of the map $$ A\mapsto \begin{pmatrix} A & & \\ & \ddots & \\ & & A \end{pmatrix} \in B(H^{(n)}). $$ So $M_n(\mathcal{A})$ is the operators on $H^{(n)}$ commuting with the diagonal action of the algebra $\mathcal{A}'$, as expected. I hope this clears up the confusion.
I think it would be more clear to use lower case letters for operators and upper case letters for algebras, e.g., $a\in \mathcal{A}$.