The arrival of the more advanced cannons built on Chinese design at the siege of Ganghwa island allowed the Manchu to sink a number of Korean ships and forcing the Korean navy to pull back. Those that remained moved sailed away from the Manchu artillery to the far side of the island. Amin, having commandeered barges from upriver, thus sailed past an astonished Yoto and his bannermen to reach the island first. Amin’s objective in taking the island was to capture and either control or eliminate the king as an obstacle to his own ambitions of becoming the overlord of the peninsula. The sudden landing of Amin’s forces took Im Gyeong Eop and his garrison by surprise, but they enacted a furious defence. Until, that is, a Manchu cavalry force, landed on the north side of the island, was able to flank the defenders from behind. As the commandeered barges moved to and fro ferrying fresh invaders, the defenders were overwhelmed. On the western side of the island, a hasty evacuation of the royal family and Western faction members was underway. The king and his officials escaped by ship, but much of the royal household was captured before they could board. Amin, enraged at missing his chance to capture the king, ordered the beheading of the Queen Inryeol and the princes Sohyeon [1], Bongrim and Inpyeong, as well as most of the unlucky courtiers and hangers-on that remained.
When news of the summary execution reached Hong Taeguk, he summoned Amin and Yoto to Hanseong. In the luxury of Changgyeonggung (창경궁 | 昌慶宮) [2] palace, he berated Amin for his hubris and foolishness. Hong Taegeuk had planned to keep the royal family as hostages to force a peace, and then furthermore to keep the Joseon on a tight leash in the future. Now it seemed that continued war was inevitable, and what’s more the executions had caused an upset among the Korean populace and even among their Chinman allies. Although Amin was technically a co-ruler of equal rank with Hong Taegeuk, he found himself outnumbered: Yoto was a crony of Hong, and many of the surrounding Korean population, Chinman and Chinmyeong alike, wished him harm. He was forced to accept a loaded suggestion to relocate to the far north of the peninsula, to complete the pacification of Pyongan and to act as a check against any aggressive moves by the Ming general Mao Wenlong. This would prove to be a mistake.
Im Gyeong Eop had, fortunately for him, been captured by Yoto’s troops and was brought before Hong Taegeuk. Contrary to his expectations, he was spared execution: his defense of Ganghwa island had won the admiration of the Manchu leader. He was, instead, offered a chance to come over to the side of the Chinman. He refused, as the war was still not over, and Hong Taegeuk accepted this with some largesse. Im would be imprisoned in Hanseong, in relative comfort, until he came around to the Manchu's way of thinking. Many of his men, however, were less patriotic. A significant portion of those captured alive on Ganghwa came over to the Chinman side, though their questionable loyalty meant they were divided among the trusted Chinman troops.
Meanwhile, the Neungyanggun fled south to the island of Jindo, famous for a breed of loyal dog. Terrified and furious after hearing reports of the murder of his family and courtiers, he began to write frantic letters to the Shogun Tokugawa Iemitsu pleading for assistance. Further supplies of muskets and gunpowder were sent from Tsushima, but a formal intervention was blocked by the Ogosho Tokugawa Hidetata [3] for the time being. In any event, with much of the formerly powerful Westerner faction stuck in Namhansanseong fortress surrounded by Jirgalang’s bannermen and Chinman forces, and much of the remainder killed by Amin’s bannermen on Ganghwa island, the political influence of the king himself had diminished.
Most of the organized Korean resistance was based around the city of Jeonju, where the Minister of War, Jang Man (장만 | 張晩), dominated. As most of the surviving government was cowering in Namhansanseong fortress surrounded by the enemy, what was essentially a military government formed. The defenders realized that in order to defeat the invaders, it would be important to utilize naval power as much as practically possible. The Korean navy was advanced, having defeated the Japanese a generation earlier. Though they lacked a naval commander with the brilliance of Yi Sunshin, there was no Manchu naval force to challenge the Korean navy. Logistically and tactically the sea was the Joseon's greatest asset.
In Hamgyeong-do, Dasen’s main rival was the Korean general Yi Rip (이립 | 李立). Despite access to additional Manchu reinforcements from across the Tumen river, Dasen was unable to strike a fatal blow against the canny general. Korean irregular forces, the mountainous landscape and Korean naval control of the East Sea conspired against the invaders. However, as time passed the Korean forces were pushed back and forced to pull back to the safety of the Yukjin (육진| 六鎭), the garrisons established along the Tumen in the 15th century to safeguard against northern barbarian aggression. With aggression coming from both sides, however, these garrisons fell one by one. Though Korean irregular attacks continued, the bulk of the organized Korean resistance pulled south to Gangwon-do by land or via sea.
[1] I feel bad for young Sohyeon, but he's an interesting and tragic figure. In OTL, he was kept as a hostage by the Qing and ended up being exposed to Catholicism by Jesuits in Beijing. He is said to have had ambitions to bring Catholicism and Western ideas to Kore, but this ruffled way too many conservative feathers, including those of his father, the odious Injo (the Neungyanggun ITTL). He ended up dying of a "mysterious" head injury in the palace. I think a "Prince Sohyeon becomes king, 17th century Korea is opened to Catholic missionaries and Western trade, hilarity ensues" timeline has a lot of potential...
[2] Changdeokgung would have been a better palace to take over, but the Neungyanggun [Injo]'s coup resulted in arson and I doubt it was completely reconstructed by this point.
[3] The position of "Retired Shogun" still meant he had control of most of the power at this point.