Kansas
According to Google Maps - the halfway point would be somewhere close to Toledo, OH
According to MeetWays, halfway between Cheyenne, Wyoming and Wichita, Kansas would be a point near Brewster, Kansas on I70.
Hillsdale, IL
Des Moines, IA
Kansas City, Missouri.
Salina Kansas
The halfway point between Kansas City, MO, and Rochester, MN, along Interstate 35 is near the town of Des Moines, Iowa. This city is roughly equidistant from both locations and offers various amenities and services for travelers. It is approximately 200 miles from both Kansas City and Rochester, making it a convenient stop for rest and refueling.
The halfway point between Melbourne and London is Akyab, Myanmar.
If you're on I-80, it's Des Moines. If you're on I-70, it's Kansas City.
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