The main reason would be that the Korean Provisional Government based in China was the recognized government of Korea. The government in exile was founded in 1919 with Syngman Rhee as the first president. It was made up of people from all over Korea, north and south, and it was the government that negotiated with the Allies. After Pearl Harbor, KPG members went to Washington to point out that Koreans were NOT Japanese and that they shouldn't be interned. Syngman Rhee spent a good portion of the 20's and 30's lobbying on behalf of the provisional government in Washington. When the war was finally over, Rhee was flown in on a US military plane and his group rose up over the other groups and won the elections in the South.
There were a few royal family members still around, BUT the Joseon dynasty had ended in 1894-95. The Korean Empire was founded, but the Japanese assassinated the Queen and were meddling so much that there was no real push to bring the royal family back. Further, many members of the royal family had either supported the Japanese, married into the Japanese royal family, and/or fought for the Japanese Imperial Army. The Allies wouldn't have restored a royal family that supported the Japanese both tacitly and openly after 35-50 years.
While many houses were "restored" to rule in the aftermath of WWI, were there examples where this was done after WWII? Around World War I, in some places (I'm thinking specifically of the Middle East and to some extent the Balkans), royal houses were sometimes put on the throne even when they didn't have historical ties to the lands that they ruled. However, the same thing did not happen after WWII and even many monarchies that were in power at the start of the war were abolished by 1950, particularly those that ended up in the Soviet sphere of influence, but monarchs that were seen as close to the Fascists were also in danger of abolition.
Looking at Europe, 1930 had far more monarchs than 1950. Indeed, besides those monarchs who went into exile with the allies, the only monarch who lost power before WWII and regained power after WWII I can think of is the King of Spain. Monarchy was abolished by the Republicans in 1931 and was brought back in steps by Franco, but this was mostly unrelated to WWII. The Korean house had been out of power for several decades by the start of the war.
As Soviet satellites, Bulgaria, Hungary and Romania all abolished their monarchies after shortly WWII. Funnily enough, some of these former royal houses (as commoners) have begun participating in post-Communist public life. Notably, Simeon Saxe-Coburg-Gotha was Tsar/King of Bulgaria 1943-1946 (he was only born in 1937 so he had little power, obviously) and then was elected prime minster 2001-2005! Not to venture too far into Bulgarian politics past the 20 year line, but he was not commonly seen as a successful prime minister and he retired from politics in 2009 when his party won no seats. The former ruling houses of Romania and Hungary The former ruling house of Romania has also returned after communism as public figures, but not explicitly political ones and certainly not as monarchs.
Albania's story is somewhat similar, except they unambiguously lost their monarchy at the start of the war (in 1939, the Italians invaded and voted Victor Emanuel, already King of Italy, as King of Albania as well) and unsurprisingly the communists did not return former King Zog (an interesting figure even by Balkan standards) to the throne. Yugoslavia's king fled during the War, Yugoslavia was broken up, the King deposed, and when it was put back together under communist rule, there was no iniative to restore a monarch. Contrast this with Greece, where the monarch fled during the course of the War, and came back shortly after the war when the first elections brought pro-monarchist parties to power (the monarchy in Greece was abolished in the 1970's after a coup; the monarch had been in exile since the coup came to power in 67).
The climate after WWII was just very different from the climate after WWI, when royal houses were restored and set up almost willy-nilly by the victors and their allies. Read sometime how Jordan ended up with a king after WWI--in short, Abdullah's brother Faisal was supposed to be king of Syria, but they got kicked out, and Abdullah basically showed up in Jordan with three hundred armed men, declared himself Emir of the British mandate, and the British basically just accepted this; Abdullah had previously been proclaimed King of Iraq, but rejected it so that Faisal, who'd just been kicked out of Syria, could be king there.
After WW2, on the other hand, even Britain kept power from the former princes who ruled princely states within in Colonial India as it moved towards independence and partition. If the country ended up behind the Iron Curtain, not only were royal houses not returned, but existing monarchies were abolished. The Westen allies didn't fight this even with monarchs who they had been (relatively) close to, as in Yugoslavia.
If the country ended up in Western influence, the exile monarchs were only brought back to some standing if the government wanted it, as with Greece in '46 and Spain later. The Netherlands, Luxembourg, and Norway had royal houses that went into exile during the War as well and were welcomed back. That is, it was treated as a domestic issue, not an interenational one. Post-fascist Italy, for instance, held a referendum and ended the monarchy in 1946. The Danish King never left, but avoided much association with the fascists (indeed, he was if anything seen as somewhat anti-Fascist, as demonstrated by the telegram crisis). The Belgian King Leopold III, also never left, but had hot and cold relationships both with the Fascists and with the government in exile. His personal post-War status was unclear, though the royal house seems to have been supported more generally--his brother severed as Prince Regent until a plebsicite allowed Leopold III to return (Belgium is and was a very divided country and, in Walloonia where he was least popular, his return was treated with protests and strikes and so he almost immediate announced that he was abdicating in favor of his son). So outside the Soviet sphere of influence, it seems very much up to domestic political actors what the role of the royal families (both those who stayed and those who went in exile) would be.
Now, I don't know much about Korea specifically, but judging from what we see in Europe we can see a few things. First, there would be obvious Communist opposition to the return of royal houses everywhere, and considering the allies didn't push for it anywhere else that Communists controlled, even with friendly monarchs (the former King of Yugoslavia was commissioned in the British Royal Air Force) who had actually been in power at the start of the war, there's no reason to expect for them to push for it over Communists opposition. This may have been a big factor in partitioned Korea, where the North was a Soviet satellite and the the allied powers in the South did not want to antagonize the North.
Second, if the royal house was seen as a fascist collaborators (which the little I've read suggests they might have been seen as such, considering they received Japanese titles and what not), there's no reason to expect that they would be restored and a decent possibly that this alone might be enough to abolish them, cf. the Italian monarchy and to a degree Belgium.
Third, royal houses were only really restored if there was domestic political support for their restoration, cf. Greece and Spain, as well as Denmark, the Netherlands, Luxembourg, and Norway and places like that. This doesn't seem to be the case in Korea. Yi Kang, the head of the once ruling house of Korea, apparently visited the Shanghai-based Provisional Government of Korea in 1919 and found little support for his claim there, according to Wikipedia. Many of the Provisional Government people obviously later became important in South Korea politics. It seems they carried their apathy towards the royal house with them when they returned to power.
In short, there seemed to be no one pushing for royal houses broadly on the international scale after WWII as there had been after WWI. These things were left up to domestic politics. Domestically in Korea, it seems like there was clear opposition to restoring the royal house and, as far as I can see in a quick glance through, little political support.
What happened to the Korean royal family after this? Do they have known descendants?
Why would they have? Sorry if this doesn't follow sub guidelines, I'm just curious about your thought process and that would help in formulating an answer.