= Solution
Substituting the two differential coordinate transformations cancels every coefficient singular as $\Delta^{-1}$, so the ingoing Kerr-like coordinates are regular at $r=r_+$. The inverse metric applied to the normal covector $dr$ has
$$
g^{ab}(dr)_b
=\frac{r^2+a^2}{\Sigma}\,\partial_v^a
+\frac{a}{\Sigma}\,\partial_\chi^a
+\frac{\Delta}{\Sigma}\,\partial_r^a.
$$
Its norm is $g^{rr}=\Delta/\Sigma$, which vanishes at $r=r_+$. Thus that level set is a <null hypersurface>. On it the raised normal is proportional to
$$
\partial_v+\frac{a}{r_+^2+a^2}\partial_\chi.
$$
Since the transformed stationary and axial Killing fields are $K=\partial_v$ and $m=\partial_\chi$, the horizon is a <Killing horizon> generated by
$$
\boxed{\xi=K+\Omega_Hm,
\qquad
\Omega_H=\frac{a}{r_+^2+a^2}}.
$$
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