Many Westerners are aware of Japanese culture, far fewer know much of Korean
This; Japan has been a world power for at least a century, and it's been exporting popular culture to the rest of the world since the last half century. (South) Korea has only recently reached the kind of cultural relevance to the rest of the world that Japan had decades ago, and it's mostly because of a conscious effort by the local government, that basically propped up a whole export-oriented entertainment industry.
Maybe if Korea had been able to modernize in the late 19th century as Japan did, elements of the local culture would've become more widely known in the West much earlier than in the 2000s; a likely POD would be the assassination attempt on Queen Min being discovered; that woman could've become Joseon's own Emperor Meiji despite not being the actual ruler of her country and, with Japan's reputation taking a hit after the assassination attempt is revealed, Japan might be forced to direct their imperialistic aims somewhere else.
Min's Korea (maybe it'd keep its old name of Joseon, since the Empire of Korea was proclaimed
after her assassination) would be aligned to Russia, allowing for an even greater penetration of Korean culture in certain areas of the old Russian Empire like central Asia, and knowledge of Korean poetry might spread from there to Moscow and Saint Petersburg and from there to Europe.
Joseon might try to take advantage of China's weakness by annexing
the region of southern Manchuria with a historical presence of Korean speakers, too, justifying the land grab by claiming it as part of
the old kingdom of Goguryeo; the capital might be moved to the former Goguryeo site of Gungnae, on the border between China and Korea as a power move - the city would have to be rebuilt almost from scratch, and it would have a somewhat Russian feel to it.
With the fall of the Russian Empire, Joseon would become a popular destination for White Russian exiles; I'm murdering a lot of butterflies here but, if World War II still happens and Joseon joins the Allies, Japan might end up losing even more than it did in OTL, with the Kingdom of Ryukyu being restored as a
Joseon satellite state.
In ATL 2017, the Kingdom of Joseon has been a prosperous constitutional monarchy for more than half a century; since it wasn't invaded and occupied by either Japan or the United States, it has retained much more of its historical architecture and culture than in OTL; also, Buddhism wasn't discredited by its association with Japanese rule, so Christianity is far less prevalent, while also being mostly Orthodox. It's not the cut-throat capitalist Hell Joseon of today, due to less American influence, but a social democratic state - its peculiar obsession with academic performance and its undercurrent of social conservatism would however remain, since they are far more Confucian than American or Christian in nature.
Long story short, it'd be a model Christian Democrat state despite not being Christian at all.
And this is how
sijo could become the most widely known form of Asian poetry in the West.
