= Solution
The two initial <Kerr black holes> have total area
$$
A_i=16\pi\left(M^2+\sqrt{M^4-J^2}\right),
$$
whereas the final <Schwarzschild black hole> has $A_f=16\pi M'^2$. <Hawking's area theorem> implies
$$
M'^2\geq M^2+\sqrt{M^4-J^2}.
$$
Since $\eta=1-M'/(2M)$, the radiated fraction is bounded by
$$
\boxed{\eta\leq
1-\frac12\sqrt{1+\sqrt{1-\left(\frac{J}{M^2}\right)^2}}}.
$$
This upper bound increases with $|J|/M^2$ and reaches
$$
\boxed{\eta_{\rm max}=\frac12}
$$
for two initially extremal holes, $|J|=M^2$. For the final hole to be Schwarzschild, their spins must be oppositely directed so the total angular momentum vanishes. The bound assumes an idealized merger saturating the area law.
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