As I [mentioned](https://mathoverflow.net/questions/431460/particular-reduced-expression-of-the-longest-element-of-weyl-group#comment1110552_431460) in a comment, this is a special case of finding a distinguished representative for a coset of a parabolic subgroup.  Let's inductively define elements $w_n$, with the convenient starting point $w_n = w_0$ when $n = 0$.  Having defined $w_n$ in general, one of two things can happen:  either $w_n^{-1}K$ consists of positive roots, or there is some $\alpha \in K$ such that $w_n^{-1}\alpha$ is negative.  In the former case, we stop, and put $v_K = w_n$.  In the latter case, we put $w_{n + 1} = s_\alpha w_n$.  It is part of the general theory of length in Coxeter groups that $\ell(w_{n + 1}) = \ell(w_n) - 1$, and hence that $\ell(w_n) = \ell(w_0) - n$ for all natural numbers $n$ such that $w_n$ is defined.

Suppose that $w^K \mathrel{:=} w_0 v_K^{-1}$ is not the long element $w_0^K$ of the parabolic subgroup $W_K$ of the Weyl group.  Then $\ell(w_0^K)$ is strictly greater than $\ell(w^K)$; but  then $\ell(w_0^K v_K)$, which equals $\ell(w_0^K) + \ell(v_K)$ by Proposition 1.10(c) of [Humphreys - Reflection groups …](https://doi.org/10.1017/CBO9780511623646), is strictly larger than $\ell(w^K) + \ell(v_K) = \ell(w^K v_K) = \ell(w_0)$.  This is a contradiction.