Whittaker vectors in singular Whittaker modules

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Borel subalgebra $\mathfrak{b}$ and corresponding nilradical $\mathfrak{n}$. We show that singular Whittaker modules $M$ are simple if and only if the space $\text{Wh}M$ of Whittaker vectors is $1$-dimensional. For arbitrary locally $\mathfrak{n}$-finite $\mathfrak{g}$-modules $V$, an immediate corollary is that the dimension of $\text{Wh}V$ is bounded by the composition length of $V$.