Pick distinct complex numbers $\lambda_1,\ldots,\lambda_k$ and consider the element
$$
X:=\Bigg(\,\underbrace{\begin{smallmatrix}
\lambda_1&1&0&0&0\\
0&\lambda_1&1&0&0\\
0&0&\ddots&\ddots&0\\
0&0&0&\lambda_1&1\\
0&0&0&0&\lambda_1\\
\end{smallmatrix}}_{n_1}\,\Bigg)
\oplus
\Bigg(\,\underbrace{\begin{smallmatrix}
\lambda_2&1&0&0&0\\
0&\lambda_2&1&0&0\\
0&0&\ddots&\ddots&0\\
0&0&0&\lambda_2&1\\
0&0&0&0&\lambda_2\\
\end{smallmatrix}}_{n_2}\,\Bigg)
\oplus
\cdots\oplus
\Bigg(\,\underbrace{\begin{smallmatrix}
\lambda_k&1&0&0&0\\
0&\lambda_k&1&0&0\\
0&0&\ddots&\ddots&0\\
0&0&0&\lambda_k&1\\
0&0&0&0&\lambda_k\\
\end{smallmatrix}}_{n_k}\,\Bigg)
$$
I claim that this element generates your algebra.

First of all, $(X-\lambda_i)^{n_i}$ has zero for its $i$th component.
Taking products of such elements, one can achieve something whose only non-zero entry is in a given summand. The remaining term is upper triangular with non-zero entries on the diagonal.

Then it's a matter of playing around to see that this generates everything.