Because we all know "any idea can be written well" is my mantra. There are three options for this, in increasing order of difficulty.
1/ Don't play it straight. This is what "My Dwimmerlaik" does, writing what seems to be solid humour at 40K word length. The names later on make my eye twitch (the Witch-King appears to be called "Eric"), but that's part of the writer's goal, I think.
2/ Set it earlier. Way earlier. The Witch-King was a Lord of Numenor (or possibly a king, or maybe even a queen), so you could write a grimly tragic romance where the central conflict is between his love for his lady and for the Ring. (Given the existence of the Tale of Aldarion and Erendis, this is very on-theme for Numenor.) Simple enough to do, because he's not undead yet.
3/ Think very hard, then set it later, in the time of Angmar or Morgul. At this point, the Nine Rings are probably held by Sauron, to ensure the Nazgul's loyalty to him. So the Witch-King doesn't have physical possession of the thing.
The #1 question you need to answer is: what, if anything, can get a Ringbearer to pay attention to something other than their Ring? The books give us a few possible answers.
a) Survival. Gollum lost the One Ring because he was more focussed on his food, and didn't try to steal it back from Frodo because he was afraid for his life. We know the Nazgul aren't in direct mental contact with Sauron, because if they were the months Frodo spent in Rivendell were plenty long enough for Sauron to get something in place to trap him. So if the Witch-King can be captured (if!), he might be persuaded to cooperate in exchange for his life.
b) Loyalty. Sam gives the Ring back to Frodo, and Bilbo leaves it behind at Gandalf's request. Even Smeagol seems to be loyal to Frodo, as far as he can. But what would the Witch-King be loyal to, other than Sauron? My answer is to go back to an older loyalty: create a character to be the Lost Queen of Numenor. She would be a Black Numenorean, probably from Umbar or its exiles, and crucially would be a direct descendent of one of the last few Kings of Numenor (which Aragorn is not). Let's say her last royal ancestor was Ar-Zimrathôn, great-great grandfather of Ar-Pharazon; his regnal name means "Collector of Jewels", which could refer to having many children. It also puts the split far enough back that her ancestors wouldn't immediately emerge as rulers of the Black Numenoreans. Perhaps she's a messenger from Umbar who recalls an old family legend about the Witch-King's identity. Or, if you wanted a canon character, she could be Queen Beruthiel. (That would be some 500 years before the Nazgul officially reappeared, so perhaps like the Orcs he was 'free' while Sauron was too weak to exert control.)
c) Mental influence from a greater Power. The Nazgul surrendered their Rings to Sauron, and Frodo lent his to Bombadil. A Maia could probably get the Witch-King's attention, and maybe a High Elf, if you can write around the whole 'burning Unseen light that causes great pain' aspect. Galadriel or Elrond could almost certainly do it, with their own Rings. Oh, which brings to mind the last option:
d) The One Ring. Find a suitably bossy Hobbit-lass (possibly Pearl Took the political assassin), somehow get the Ring into her hands, and have her command the Ringwraiths' obedience. There is nothing humourous about this concept at all. >:(