Since $L/K$ is finite, we can write $L = K(a_1,\ldots,a_r)$ for some $a_i$. These same $a_i$ will generate $L_w$ over $K_v$. You can't hope for anything simpler than that! Well, you could hope that a $K$-basis of $L$ is a $K_v$-basis of $L_w$, but that is WRONG in general. For instance, $K_v$ can sometimes equal $L_w$ even if $[L:K] > 1$, e.g., the $(1+2i)$-adic completion of $\mathbf Q(i)$ is the $5$-adic completion of $\mathbf Q$.

To prove $L_w = K_v(a_1,\ldots,a_r)$, one containment is easy: since $K_v$ and each $a_i$ is in $L_w$, we get $K_v(a_1,\ldots, a_r) \subset L_w$.

Since $K$ is in $K_v$ we get $L = K(a_1,\ldots,a_r) \subset K_v(a_1,\ldots,a_r)$. Thus
$$
L \subset K_v(a_1,\ldots,a_r) \subset L_w.
$$
The $w$-completion of $L$ is $L_w$, so
$$
(K_v(a_1,\ldots,a_r))_w = L_w,
$$
where the field on the left is the $w$-completion (or closure) of $K_v(a_1,\ldots,a_r)$ inside $L_w$. We are NOT done because we need to explain why the $w$-completion of $K_v(a_1,\ldots,a_r)$ is itself, in other words why this field is complete with respect to the absolute value $w$ on $L_w$. We will use a theorem about extending absolute values to finite extensions of a complete field.

Theorem. If $F$ is a complete valued field then any finite extension of $F$ admits a unique extension of the absolute value on $F$ and it is complete with respect to that extension.

You can find a proof of the unique extension of absolute values from a complete field to a finite extension in Cassels's book Local Fields. It is also in Gouvea's book p-Adic Numbers: An Introduction. Koblitz's book on p-adic numbers and zeta-functions has a proof that assumes the base field is locally compact, but the other proofs avoid such a hypothesis.

We can apply this theorem because $K_v(a_1,\ldots,a_r)$ is a finite extension of $K_v$: a nonzero polynomial in $K[x]$ with $a_i$ as a root also has coefficients in $K_v$, so each $a_i$ is algebraic over $K_v$. Thus $K_v(a_1,\ldots,a_r)$ is a finite extension of $K_v$ and the absolute value $w$ on it from $L_w$ extends the absolute value $v$ on $K_v$, so the theorem implies that $K_v(a_1,\ldots,a_r)$ is complete with respect to $w$. QED

Neukirch, Algebraic Number Theory, but I'm not sure if it's really trivial or not. $\endgroup$