The PoD is that the politicians decide that Admiral Lee Soon Shin, perhaps the best admiral in the world at the time, isn't a threat to them politically so the politicians let Admiral Lee do whatever he wants (in OTL, the politicians are afraid of his popularity and jealous of his victories, that they force him to become the rank of a common soldier before re-promoting him). Additionally, since Lee Soon Shin rekt the Japanese navy early on, he never fights the Battle of Sachon Bay, the battle that broke Japan's back for a long time. As a result, Admiral Lee never dies and continues to live on.
Could the history of Korea be changed from this one man? What will be the effect of having the great Admiral Lee alive for around 20-30 more years? Can he participate in politics with the support of the people? Would he lead an invasion force of Japan? How could history differ from this one small change?