I have to admit it looks to me, from everything I've ever heard about Japanese society and history, that in Korea at least Japan is going to be on a collision course with deep-seated Korean pride and identity. The problem here is that the Japanese, in the most liberal and tolerant configuration imaginable, are still going to assume their own innate superiority and see Korea at best through the lens of "progress=becoming more Japanese." The Koreans won't go for it.
The best evidence we have of the greater Empire having some potential for stability is Taiwan; the Japanese regime there did experience some serious unrest but compared to the usual reputation they picked up in conquering still larger swathes of Asia, the Taiwanese did by and large make their peace and were an integral and actively patriotic part of the Empire during WWII. The thing is, that peace was again very largely on the terms of the Taiwanese accepting increasing integration into Japanese norms. In linguistic and cultural terms, Japanese was becoming increasingly dominant, and in political terms Taiwan was becoming more represented via members of the Diet in Tokyo being admitted from Taiwan.
Korea would be different; the best evidence we have is that Japan would at best try the same tactics of "acceptance via assimilation" that were working for them in Taiwan, but these tactics would show much less sign of "progress" from a Japanese point of view.
To hold Korea without the need for ongoing overt oppression, Japan needs to come to terms with the notion that the Koreans are a different people with their own history and dignity and they aren't going to merge smoothly and submissively into Japanese culture.
All this said
I know of at least one unfortunately short-lived and fragmentary timeline that simply assumed the results the OP wanted to see, and it was quite interesting. Of course part of the interest for me was trying to see how the author proposed to explain the Empire lasting as a stable entity into the 2000s and beyond, and unfortunately the author seems to have dropped it without getting around to these explanations.
It wasn't a normal timeline story either; it was about aircraft getting exchanged in a sort of coupled ISOT incident between our timeline and theirs, so a jet from the Imperial timeline lands at OTL Vancouver, and shenanigans ensue. We learned that Taiwan and Korea were indeed still in this Empire and that it didn't extend much if at all beyond the bounds of OTL 1930 or so, either they never got into any later wars with major powers or these wars ended without dramatic changes to the Empire, one way or the other.
So I was left to speculate how and why this comes about.
The best I could do, and thus the best I can recommend here, is that the POD is a subtle one in Meiji times, having to do with the exact nature of the Constitution adopted which strengthens the hands of liberals and somewhat restrains the tendency toward militarization. But I don't know to what extent stacking the deck in favor of moderation would, if successful, mean the Japanese refrain from their efforts to acquire control of Taiwan and Korea in the first place! I suppose that's why someone upthread suggested a later reform--things go on OTL tracks until after these have been taken and then someone sees the light. But if there's to be hope that restraint will prevail, I suspect the institutional checks have to be pretty deeply rooted.
And to have some sort of peace in Korea, something needs to have happened to knock a sense of respectful tolerance of diversity into Japanese heads, and to conciliate Korean pride and dignity to accepting continuned Japanese leadership.
Is there any reasonable way the initial entry of Korea into the Empire can be by a more mutual process? Can our "Taisho" Japan actually be some kind of dual monarchy?