= Solution
Let
$$
T=\{x_1,\ldots,x_k,y_1,\ldots,y_k\}
$$
be the terminal set, and let $C$ be the vertex set of the given <complete graph> $K_{2k}$. Any vertex set $S$ meeting every $T$-$C$ path has size at least $2k$: if $|S|<2k$, choose $t\in T\setminus S$ and $c\in C\setminus S$. Since $G$ is $2k$-connected, $G-S$ is connected, so it contains a $t$-$c$ path avoiding $S$, a contradiction.
The <Set version of Menger theorem> therefore gives $2k$ pairwise vertex-disjoint $T$-$C$ paths. Truncate them at their first and last vertices in $T\cup C$. Since both sets have $2k$ vertices, every terminal is the endpoint of one path and the other endpoints are distinct vertices of $C$. Denote the path from a terminal $z$ to its clique endpoint by $R_z$, and call that endpoint $c_z$.
For each $i$, the clique contains the edge $c_{x_i}c_{y_i}$. Concatenating
$$
R_{x_i},\quad c_{x_i}c_{y_i},\quad R_{y_i}
$$
gives an $x_i$-$y_i$ <path in a graph>. The linkage paths are mutually vertex-disjoint, their clique endpoints are all distinct, and the joining clique edges pair those endpoints without introducing a new vertex. The resulting paths $P_1,\ldots,P_k$ are therefore mutually vertex-disjoint. In particular, under the stated clique hypothesis, the $2k$-connected graph is <linked graph>[$k$-linked] for these terminals.
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