= Solution
The <Killing equation> implies
$$
K^a\nabla_aK^b=-\frac12\nabla^b(K^aK_a).
$$
Here $K^2=-\lambda^2$, so in the regular coordinates
$$
K^a\nabla_aK^b
=\lambda\,g^{b\lambda}
=-\frac{\lambda}{M}(\partial_v)^b
+\frac{\lambda^3}{M^2}(\partial_\lambda)^b.
$$
This vector vanishes at $\lambda=0$. The <Killing vector field> $K=\partial_v$ is nonzero there, so comparison with the definition of <surface gravity> yields
$$
\boxed{\kappa=0.}
$$
Hence this is a <degenerate Killing horizon>. The argument uses the vanishing gradient of $K^2$ on the horizon; the condition $K^2=0$ by itself would not force zero <surface gravity>.
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