A couple things worth noting:
1. Hideyoshi is still likely going to die before his heir comes of age, considering he was 62 and his son Hideyori was 5 in 1598. Tokugawa Ieyasu still has great ambitions and the Hideyoshi supporters in Japan, rather than being strengthened by the conquests, will probably need years for their gambit in Korea to actually pay off. Not only will the peninsula's arable lands still be devastated due to the war itself, the primary beneficiaries of the war will need to garrison their new holdings with a not inconsiderable percent of their forces due to guerilla activities. Sekigahara was only 2 years after the Imjin War ended, so the Tokugawa becoming ascendant is not butterflied away.
2. The Ming initially dragged its feet because the Ming court was suspecting the Joseon were collaborating with the Japanese and faking an invasion, based on how quickly the peninsula fell. Hideyoshi's endgoal being the invasion of China itself would likely be enough to get the Ming to commit more heavily in the war than OTL, though supply lines are going to be the main bottleneck. That said, while supplies were a major issue, the internal squabbling between Japanese generals about who would get more glory and their tendency to march faster their their overland supply lines could be established were also a major factor in both the Japanese defeat and the eventual defeat of the Toyotomi clan at Sekigahara.
3. Speaking of supplies, Admiral Yi was vital is cutting off the overseas supply lines for the Japanese, but his absence does not butterfly away the presence of volunteer militias (Righteous Armies) that would harry the Japanese on the peninsula itself or the farmlands being ruined or abandoned during the war. Japanese forces were reported to be extremely brutal with their massacres and the rape of Korean women by Japanese soldiers was a rallying call for the peasantry to rise up. Add to that the monks in their mountain temples and the general terrain of Korea itself (forested mountains and hills everywhere you look) that the locals know from birth and the Japanese will be unfamiliar with, and that's years of unrest that will be difficult to pacify, if ever. The Righteous Armies also co-opted the army's artillery supplies, apparently, so it would be more difficult to gain total control.
4. During negotiations with the Ming, the Ming would not brook the thought of considering Japan an equal. Remember, this is the court that, even when their own armies of tens and hundreds of thousands were getting swept aside by the Later Jin, later Qing, forces, refused to compromise with Hong Taiji and simply give tribute. Hideyoshi is even more brazen than that, demanding not only a Ming princess and recognition of imperial stature the Ming refused to give the Manchu even after incredibly brutal defeats.
5. Hideyoshi only demanded half of Korea probably because he wasn't actively winning anymore. If Japanese forces are doing better than OTL, why would Hideyoshi, the man who intends on usurping the Ming emperor, settle for what he demanded when his final victory was far harder to attain? And that is in direct conflict with 4., so a negotiated peace is not an option, not without a great deal more conflict. Regardless of the situation, you have two parties that lack humility as much as they have bodies to throw into the meatgrinder. Regardless, I would say a north south split is not in cards in this situation. You'd need Hideyoshi tempering his ambitions (which would happen if the war is not going well for Japan) and the Ming getting exhausted much earlier (bit mutually exclusive, those two) to have that happen.
6. Apparently the Japanese also invaded the Jurchen during this whole mess and likely don't intend on playing nice with them, considering Hideyoshi's megalomania and Nurhaci's own ambitions. Heaven cannot brook two suns, after all, and the Jurchen and Japanese cannot both conquer China. And only one of them reliably recruited Ming turncoats and co-opted Ming bureaucracy.
TLDR: Regardless of who wins the war on the field, the Ming have been bullheaded enough to refuse peace and keep fighting with even worse losses rather than recognize another nation as of equal stature. The Japanese are not going to settle on half the peninsula if they seem like they're winning more easily than in OTL, not being led by a megalomaniac like Toyotomi Hideyoshi, so no partition. The Korean countryside is not going to be pacified after the initial wave of brutality, nor will the fields be enough to sustain the locals, let alone the invaders, and as such the Japanese are not going to see returns on their conquest for at least a few decades. Japan is also not going to be stable for long after the war after Toyotomi Hideyoshi dies and leaves behind a child under the ambitious watch of Tokugawa Ieyasu.
Based on all that, I'd say the most likely outcome is that the Ming, after reaching their logistical limits, withdraw to a more defensible border and sign a ceasefire that doesn't acknowledge Japanese gains or gives Hideyoshi what he wants. They'll keep funding the Righteous Armies to harry the Japanese relentlessly. The Japanese daimyos are going to squabble over their gains while also having to put down rebellions constantly, which will drain their forces. I don't believe Admiral Yi's existence is going to alter Hideyoshi's death all that much seeing they weren't within 500 km of each other, so, assuming he dies on schedule, Tokugawa Ieyasu has an easy time rolling over the Toyotomi forces at Sekigahara, now split between Korea and Japan. And at that point, we have rebel daimyos in Japan refusing to submit to the Tokugawa, a vengeful Ming, a turbulent Korean countryside full of rebels, and Jurchen tribes in the process of unifying.
