There's no need to go the northern route, since Korea was much closer, and easier on the logistics.
There were many factors contributing to Mongolian's loss. Being a land-based military, they had little experience with ocean vessels, and constructed a fleet of Chinese riverboats. The Koreans were quite horrified and told them that they should use ocean-going ships with curved keel, but the Mongols simply ddin't understand or didn't care. So when the typhoon came, *poof*.
The Japanese got smart after the first invasion, and built strong fortifications along potential landing points. These fortitications allowed the numerically and technologically inferior Japanese to defend against a larger and stronger invasion force, until the typhoon came and blew the Mongol fleet away.
Also, the Mongols had to go through Korea to reach Japan, and they took way too long to conquer Korea. Their first 6 major campaigns lasted almsot 30 years (1231-1259), and the final resistance didn't end until the Sambyeolcho rebellion was put down around 1271-1273.
This means that the invasion of Japan was much-delayed, and the Japanese had plenty of warning to fortify Kyushu. Had the Mongols been able to invade earlier, the first Battle of Hakata Bay would've prolly turned into a complete Japanese rout, making the second invasion unncessary.