Complementary $T$-invariant subspace of some subspace of dimension of degree of nilpotency of the nilpotent operator $T$

Question

Let $V$ be a vector space of dimension $n$ over some field $F$, let $T : V \to V$ be a nilpotent linear operator of nilpotency degree $k <n$. Then $\exists v \in V$ such that $T^{k-1}(v) \ne 0$ so that $W:=\operatorname{span} \{v,T(v),...,T^{k-1}(v)\}$ is a subspace of dimension $k$.
My question is: does there exist a complementary subspace $U$ of $W$ such that $U$ is $T$-invariant ?

EDIT : I would appreciate an argument without using the Jordan form.