The British blood that was shed for Japan must be repaid
How and where? The moment Britain declares for Japan Russia is screwed with no way out. There's no way for them to reinforce their holdings and their fleet can't stand up to the British and NO ONE in Europe is going to be on their side here. They seek peace and Britain pushes (very little pressure needed as Japan knew it could not sustain a war against Russia) Japan into agreeing to a peace. Not a drop of British blood is shed so no interest in Korea.
The government cannot leave war without gain (and Korea will be the proper reward for Britain)
Governments and especially the British one have been going to war for little or no gain since the middle of the 19th century it's all part of the 'great game' at this point. There's literally nothing for Britain to gain here and no incentive to push for making Russia more nervous and pissing off Japan who entered this whole conflict OVER Korea.
Sakhalin, no bankruptcy, big compensation for the British, more than enough for Japan (and British Korea is a plus for telling people they've won something))
No one in Britain would care about Korea, they hardly cared about the Empire they had outside of certain long-term holdings. Africa was already proving to be a drain instead of a gain and Korea would be worlds worse all around. Nobody in Britain wanted it and they'd be happy to leave it to Japan as a counter to Russian influence in the region.
Russia was terrified of active British intervention and everyone understood that the moment Britain decided to move (or not) would be the deciding factor of the war.
Randy