Most of my experience with the Great Plains comes from the Texas panhandle, but I've also done four long tours in the American West. Kansas, 'fraid not, but that's about 40 miles from the panhandle, with nothing in between that can block the wind. Close enough.

In the western states, the wind will almost always come out of the west or southwest in late spring and summer. As mentioned already, it picks up throughout the day, so you've got two reasons to wake up early: avoid wind, avoid heat. And if that's not enough, starting early means finishing early, which gives you a chance to visit stores, libraries, etc. while they're still open.

On an "average" day, I'd say the wind would be roughly 5 mph in the morning, 15 mph in the afternoon. Sometimes less, sometimes more. Every so often, a lot more.

There's usually a little less wind in the middle of summer, more in spring, so you could consider later dates, but that means it's also going to be hotter. Or you could reverse direction, which would not only give you tailwind most of the time, but you'd also go downhill more often than not. Whatever your reasons for going westbound, do you consider them more important than headwind vs. tailwind, uphill vs. downhill? Up to you.