= Solution
Use the exterior-normalized <Killing vector field> $K=\partial_t=\partial_v$ from the previous part. In <Ingoing BTZ coordinates>, its only nonzero acceleration component on $r=r_+$ is
$$
K^b\nabla_bK^v=\Gamma^v{}_{vv}=\frac12f'(r_+),
\qquad K^b\nabla_bK^r=\frac12ff'\big|_{r_+}=0.
$$
The defining equation for <surface gravity>, $\nabla_KK=\kappa K$, therefore gives the positive future-branch value
$$
\boxed{\kappa=\frac{f'(r_+)}2=\frac{r_+}{\ell^2}=\frac{\sqrt M}{\ell}.}
$$
This value uses exactly the prescribed <Killing vector field> normalization. A constant rescaling of the <Killing vector field> rescales its <surface gravity>.
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