As long as the One Ring was intact, the Nazgul were effectively undying. Note that I did not write "immortal" -- they were not immortal any more than Bilbo or Gollum was immortal. Bilbo said it very well:
'I am old, Gandalf. I don't look it, but I am beginning to feel it in my heart of hearts. Well-preserved indeed!' he snorted. 'Why, I feel all thin, sort of stretched, if you know what I mean: like butter that has been scraped over too much bread. That can't be right. I need a change, or something.'
Bilbo did not gain more life by wearing the Ring, though he did gain more duration. Gandalf put it this way:
'A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings. Yes, sooner or later - later, if he is strong or well-meaning to begin with, but neither strength nor good purpose will last - sooner or later the dark power will devour him.'
The Nine did this to the humans who wore them and they became the Nazgul. (The Three were unsullied by Sauron's touch and in any event never were held by a mortal. The Seven went to the Dwarves but they proved to be resistant to most of their rings' ill effects and, basically, just became greedier.)
Now to the death of the chief of the Nazgul:
But suddenly [the head Nazgul] too stumbled forward with a cry of bitter pain, and his stroke went wide, driving into the ground. Merry's sword had stabbed him from behind, shearing through the black mantle, and passing up beneath the hauberk had pierced the sinew behind his mighty knee.
'Éowyn! Éowyn!' cried Merry. Then tottering, struggling up, with her last strength she drove her sword between crown and mantle, as the great shoulders bowed before her. The sword broke sparkling into many shards. The crown rolled away with a clang. Éowyn fell forward upon her fallen foe. But lo! the mantle and hauberk were empty. Shapeless they lay now on the ground, torn and tumbled; and a cry went up into the shuddering air, and faded to a shrill wailing, passing with the wind, a voice bodiless and thin that died, and was swallowed up, and was never heard again in that age of this world.
...
Then [Merry] looked for his sword that he had let fall; for even as he struck his blow his arm was numbed, and now he could only use his left hand. And behold! there lay his weapon, but the blade was smoking like a dry branch that has been thrust in a fire; and as he watched it, it writhed and withered and was consumed.
So passed the sword of the Barrow-downs, work of Westernesse. But glad would he have been to know its fate who wrought it slowly long ago in the North-kingdom when the Dúnedain were young, and chief among their foes was the dread realm of Angmar and its sorcerer king. No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will.
What exactly happened here is not completely clear. In the one case it appears Merry killed the Witch King: "No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will." Yet something happened a second later when Eowyn stabbed the W-K in his unseen face: "with her last strength she drove her sword between crown and mantle, as the great shoulders bowed before her. The sword broke sparkling into many shards."
Note that Aragon says of Frodo's sword thrust at the Witch-King on Weathertop:
'The only hurt that it did to [Frodo's] enemy, I fear; for it is unharmed, but all blades perish that pierce that dreadful King.
I lean towards Merry doing the deed and Eowyn's sword breaking being either her stroke breaking on a not-yet-dead W-K or it being an ancillary effect of the W-K's demise. But reasonable people do disagree on this.
Regardless, it is clear that (1) the W-K, like any human wearing a Great Ring endures as long as he posses it, but does not gain more life and turns into "butter scraped over too much toast" "until at last every minute is a weariness".
(2) Merry's barrow-blade was essential: "No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will."
(This is not to say that Merry's blade was necessarily unique -- other, like, blades may have been made by smiths of Westernesse. But it took a special blade made by a technique lost in M-E.)
(3) We don't actually know anything about how the blade was made, or by what mechanism it operated, or by what mechanism the Nine gave the Nazgul long unlife. We only know how Angmar actually died and that no other Nazgul died until the One was unmade and the Nine failed. Speculation about other ways to end Nazgul is nugatory.
(4) This is not to say that the Nazgul consequently had nothing to fear. There is no reason to believe that they could not be damaged -- short of death -- like anyone else. And not just physically:
More deadly to [the Witch-King] was the name of Elbereth.'
The Rings gave their wearers much, but not invulnerability. And even if they were invulnerable, a Nazgul who failed would have to return to Sauron and confess his failure:
Or he will not slay thee in thy turn. He will bear thee away to the houses of lamentation, beyond all darkness, where thy flesh shall be devoured, and thy shrivelled mind be left naked to the Lidless Eye.'
Being naked to the gaze of the Lidless Eye was the fate of failure for the Nazgul. And that was reason enough to fear a foe who was not afraid, since most of their power came from the miasma of fear that surrounded them. A fearless foe was in their league and in turn to be feared.
