$\kappa$ must be an affine isometry. If $\gamma:[0,1]\to\mathbb{R}^n$ is a smooth curve and $L(\gamma)$ denotes its length, then
$$
L(\kappa\circ\gamma)=\int_0^1|D(\kappa\circ\gamma)(t)|\, dt=
\int_0^1|D\kappa(\gamma(t))\gamma'(t)|\, dt=\int_0^1|\gamma'(t)|\, dt=L(\gamma).
$$
This is because $D\kappa\in O(n)$ and hence this linear map preserves lengths of vectors. The above caclulation shows that $\kappa$ preserves lengths of curves from which it easily follows that it preserves distances and hence it is an isometry of $\mathbb{R}^n$.