Both Russia and Japan would be trying to influence the others portion of Korea and boot the other one out.
Whichever side supported Emperor Gojong tho, the other would have to find another ruler to support, not sure who would fit the bill, neither of them would be interested in supporting any socialist or nationalist leaders as they would both be interested in eventually annexing Korea.
Maybe the Japs play the Pan-Asian card promising to free the north form the 'evil' Russians, probably there would be another Russo-Japanese War within a decade.