Assuming he gets involved in US politics, it depends upon his associates. Somehow I can't quite see his views meshing all that well with Wilson's; if anything, at the time of World War I, the Progressive wing of the GOP would be more to his liking. I could see him falling in with Charles Evans Hughes, perhaps. If so, possibly he, instead of Harry Daugherty, could have become Harding's AG. In turn, that might have butterflied away some or all of the Teapot Dome scandal.
Granted, that wouldn't have altered the fundamental fact that Harding was incompetent, but it might have made his tenure somewhat less absurd.
Now, if somehow Hughes' daughter had not died in (I think)late 1919, Hughes himself might have been the GOP candidate in 1920, given the recent death of TR. In that case, King could very well have been Secretary of State, and might have established himself as a sort of grey eminence for Hughes and his successor(s).