Alternatively to kill a hydra, cut off its tail.
But let's try a harder problem, like killing a were-thunderbird. You can write off any weapons made of conductive metals, since its lightning is going to melt them and electrocute you. It's safe to assume it produces magnetic fields strong enough to bend most any matter or radiation around it out of shape; even the heat and shockwave of a nuclear weapon may be deflected with magnetic fields strong enough. And this is a godlike mythological beast out of ancient America, so I wouldn't make any bets about the limitations of its power. Expect the full set of reality-warping powers of the Freakangels, if anyone remembers them.
And it's probably really pissed off at the state of the environment and the genocide of Native Americans. Heck, I'm happy I'm on another continent and I just made up this creature myself.