I rode KS-99 from Pawhuska, OK to to Eureka, KS then over to Cassoday, KS (via US-54 to Rosalia then county roads). From there, KS-177 to Alta Vista (the Flint Hills road and is very nice, even in August) and a nice county road (old K-10) to Alma. From there, I entered Nebraska at Summerfield, KS but then I was too east for your route. I can recommend my route except the stretch between Westmoreland and Blaine, KS had a ton of gravel truck traffic in mid-morning. Camping and food was available. If you want, contact me via private message and I can send you the GPS codes and a cue sheet if you want.
I have also been on US-183 between La Crosse, KS and Holdredge, NE. It was a "make miles, not smiles" route as it is a road with usually a decent shoulder but the traffic was higher than I like. It is doable but not as pleasant.
Another suggestion is to take my route up to Alta Vista, then work diagonally over to Junction City on KS-52 or Humbolt Creek Rd. From Jct City, head NW toward Clay Center via county roads and KS-15. From there, almost any road except for US-81 has sub-2,vehiclesales per day so should be good. If you are not opposed to riding on chat (finely ground gravel), you might consider taking the Cowboy Trail, a rail trail going from Norfolk, NE west to Valentine, NE. From there, head north up to the Black Hills on another rail trail, Mickelsoneson(spelling?) Trail, which is also very nice.
As a final note, I might take a tour this fall going from Greenville, MS back to Tulsa roughly in a Greenville to Russellville, AR, to Mulberry to Winslow, AR to Tulsa mainly via low traffic highways or county roads. If you want that roGPS's gps codes and a cue sheet, let me know. Actually, I have it routed down to DeFuniak Springs, FL but don't have time to do it all this fall.
I agree wholeheartedly about checking other sources such as DOT traffic counts, etc., but wanted to give you a quick answer. At least KS has a map. My state (OK) just doesn't see the need and the OK bicycle advocacy group doesn't either unfortunately but that is another issue.