A town that is roughly halfway between Kansas City, MO, and Elgin, IL, is Peoria, IL. Located along the Illinois River, Peoria is about 200 miles from both Kansas City and Elgin, making it a convenient meeting point. It's known for its vibrant downtown and various cultural attractions.
Morton, IL
Van Horn, TX
Kansas
Around/between Charleston and Arcola IL, depending on what route you take.
The halfway point between Peoria and Topsail Island is Point Pleasant, WV, on US-35 just on the West Virginia side of the Ohio state line. (Route highlights: I-74 to I-70 to US-35 to I-77 to I-74 to I-40)
According to MeetWays, halfway between Cheyenne, Wyoming and Wichita, Kansas would be a point near Brewster, Kansas on I70.
Kansas City, Missouri.
Salina Kansas
The halfway point between Melbourne and London is Akyab, Myanmar.
The halfway point is basically the area where Nebraska, Colorado and Wyoming meet. This is not an exact answer, but in relative terms this is the halfway point between Salmon, Idaho and Kansas City, Missouri. If your looking for a town, then I would say the capitol city of Wyoming (Cheyenne) is pretty much as close as your going to get to a city halfway between Salmon and K.C.
Terre Haute, IN