I know how to compute Jordan Canonical Form of a nilpotent matrix, but I find a little bit tedious and long my method for computing the JCF of a general triangulable matrix. I'll show you how I compute the JCF, if you think you know a faster way to do it, please tell me.
Let $A\in M_{n\times n}(\mathbb{R}$) be a triangulable matrix, and let $\text{Sp}(A)=\{\lambda_1, \ldots, \lambda_k\}$, with $k\le n$, its spectrum and $p_A(t)=(-1)^k(t-\lambda_1)^{h_1}\cdots (t-\lambda_k)^{h_k}$, with $h_1+\cdots+h_k=n$, its characteristic polynomial. By primary decomposition theorem $\mathbb{R}^n=\text{Ker}(A-\lambda_1I)^{h_1}\oplus\cdots\oplus\text{Ker}(A-\lambda_kI)^{h_k}$ and each addendum is $A$-invariant. Let $W_{\lambda_j}=\text{Ker}(A-\lambda_jI)^{h_j}$. We also know from the theory that $\dim W_{\lambda_j}=h_j$ and that the characteristic polynomial of the restriction of $A$ to $W_{\lambda_j}$ is $\pm (t-\lambda_{j})^{h_j}$.
Now, what I do when I compute JCF, is writing primary decomposition, finding a basis $B_j$ of $W_{\lambda_j}$ and computing $M_j=\mathcal{M}_{B_j}(A)-\lambda_j I\in M_{h_j\times h_j}(\mathbb{R})$. At this point $M_j$ is a nilpotent matrix and I'm happy.