Question: how did eowyn kill the nazgul?
Answer: When Merry stabbed the Witch-King, unbeknownst to him, he broke the spell binding the Nazgûl to Arda, which rendered him vulnerable to other attacks. In that moment, Éowyn, with the last drop of strength she had, thrusted her sword where it would be the Witch-King's face, effectively slaying the Lord of the Nazgûl.