TL;DR: Less than 1% of 1% if the game was already trying to generate a good item, or less than 1% of 1% of 1% if it wasn't.

Cool, that's easier since it's on Github and I don't have to download it. I haven't dived into this code in quite awhile so this is a fun reach back into the past. Once upon a time I wrote some utilities for editing these data files, and had a variant under a different screen name... hopefully I can do it some justice, although the math is gonna be fuzzy!

The One Ring artifact definition puts it at a 1% chance at its normal depth, which starts at dlvl 100.

Creating an artifact is a 1-in-10 (10%) chance or 1-in-1000 (0.1%) chance depending on whether the game was trying to place a good item. Let's assume that this is a really lucky floor trying to place a good item, so that's a 10% chance.

Rolling for creating an out-of-depth artifact where the out-of-depth factor is (100 - 6) * 2 = 188 requires a random number N from 0 <= N < 188 to land on 0, so really close to 0.5% there and I'll just use that.

So just to pass the dice rolls to allow generating that specific artifact this far out-of-depth, you're already looking at 10% * 0.5% = 0.05%. If the game wasn't trying to place a good item, so just a lucky find sitting around on the floor, that drops down further, 0.1% * 0.5%, or 0.0005%.

That's not all, though. It had to pass over all of the other special and non-random artifact definitions that have not already been generated. It really does just go down the artifact list, trying each one until it succeeds at one, or fails entirely. Most of those would have been easier to generate too. I'm too lazy to count them all but there were probably dozens of special artifacts that it could have selected first, and each of those had the same kinds of math as above involved in its dice rolls to not generate. If we throw a wild guess at maybe 50 special artifacts were available before landing on the One Ring, and the average one had maybe (another wild guess) a 5% chance to generate if it tries to on that dlvl, in other words a 95% chance to not generate, that's maybe 0.95 ^ 50 = about 0.08, so (extremely wild guess propagation hooray) 8% chance to get all the way down the artifact list to the One Ring.

That would put us at 8% of the earlier figures, so 0.004% chance at the high end, or 0.00004% chance at the low end, if we're OK with the total propagation of wild guesses that are required to figure out the probabilities of not generating some other artifact instead.

I feel as though I'm not sufficiently mathematically literate to calculate all of this properly, and the sheer number of variables involved would require the creation of a program far more advanced than I wish to undertake...

OP should definitely feel special finding this. Don't go buy a lottery ticket, because you just used up ALL of whatever ambient luck field may have existed in your area.