Why can't I just start with $N$ as a simple $T_n(k)$-module, and let $k$ act on $N$ via constant diagonal matrices. Won't this turn $N$ into a $k$-vector space, and $\{aI \cdot n \mid a \in k, n \in N \}$ is a nonzero submodule, so by simplicity $N = \{aI \cdot n \mid a \in k, n \in N\}$, so $N$ is $1$-dimensional...Is there something wrong with this?