As Lubin points out below, the claim in my original post (see edits) was false.
Amended Post
The book I am reading contains the following theorem:
Let $L/K$ be a finite extension of complete non-arch. fields with residue fields $k_L$ and $k_K$ resp.
Let $\alpha \in k_L$ be separable over $k_K.$
Then there exists $a \in \mathfrak{O}_L$ such that $a\equiv \alpha \text{ mod } \mathfrak{P}_L$ and
$$[K(a):K]=[k_K(\alpha):k_K].$$
In particular, $K(a)/K$ is unramified.
It's the "in particular" bit I'm stuck with. For $K(a)/K$ to be unramified, we need
$$[k_{K(a)}:k_K]\geq[k_K(\alpha):k_K]$$
which, of course(!), doesn't imply that $k_{K(a)}\cong k_K$ as I mistakenly claimed above.
It is this that I am stuck with. Many thanks!