That the KKK essentially ran Indiana in the 1920s is beyond dispute. Also beyond dispute: that it split the Democrats badly, especially in 1924. All that said, assuming D. C. Stephenson minds his manners and does not commit rape, perhaps the KKK could have taken control of Indiana through election of a slate of individuals from the governor on down.
An ad hoc alliance with more southern state organizations might then follow: perhaps sufficiently so to prevent Al Smith's nomination by the Dems in 1928. That too might have been a bad year for the Dems, but it was anyhow. By 1932, assuming Wall Street still crashes, it might be possible for a KKK-backed semi-demagogue to gain enough popular anti-Wall Street support to gain the Dem nomination (the KKK didn't have much traction with the GOP, especially in the northeast) and challenge Hoover, perhaps successfully: it may not be out of the question that Garner might have used some KKK leverage to gain the nomination.
Once in power, the KKK of the day would use its xenophobic/racist dogma to institutionalize an America First-style policy in the '30s that would probably remain on good terms with Hitler; at the same time, relations with Japan probably would have deteriorated as in OTL. Thus, one may have a "world war" which was in reality two wars: Nazi Germany (and fascist Italy FWIW) against a coalition of France, the UK and the USSR in Europe, and the US against Japan in the Pacific. Neither would have much to do with the other, unless the British were willing to make a deal with the devil for something along the lines of Lend-Lease--if indeed those in power in the US would go for something like that, which is debatable.
You wouldn't be likely to see a Manhattan Project as in OTL: too many of the best brains were foreign (anathema to the KKK) and a lot of them Jewish (doubly so). The closest analogue would have to have come about in Great Britain, which didn't have nearly the industrial (especially the chemical) capacity of the US of the time. I think that means both wars drag out somewhat more: perhaps to 1946 or 1947 in Europe; definitely to 1947 or even 1948 in the Pacific.
The US would be a lot less pleasant place: race relations would pretty much stagnate as they were about 1915 or so, especially in the south and border south. There might be areas of the nation where blacks weren't too badly off-say, New England or the Pacific Northwest-but those would be the exception to the rule. Look for a lot of minorities, if able to do so, to leave the US for Canada.
Would the KKK collapse eventually? Perhaps, if there were some sort of misstep. After all, what took down Al Capone was running afoul of income tax laws. Maybe the same might take down the KKK after the Pacific war.