Theoden was not referring to the Nazgul, but rather to the Southron king with the Serpent Banner.
I go back and forth on who slew the Nazgul Lord, though I lean towards Birdland's view not only for the same reasons per the quote she cites, but also because Eowyn's sword shatters on "impact". Aragorn, after the Weathertop incident, also seems to suggest that a normal blade will not harm a Nazgul.
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Beleriand, Beleriand,
the borders of the Elven-land.
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