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.