From eastbound I-70: Just after crossing over the border from Colorado to Kansas, take Exit 1 and head south on County Road 3. Stay on CR 3 for approximately 22 miles. You will eventually see a sign for Mt. Sunflower. Turn right (west) at the gate onto a narrow road and continue for a couple of bumpy miles. Turn right (north) at the cattle gate and a short distance to the summit marker.
As a state high-pointer on a cross-country roadtrip, I attempted to make the trek to Sunflower this morning. BE ADVISED: with any rain, the dirt roads required become impassable morasses of slick mud and gravel. Even in a well-laden four wheel drive with an experienced driver, we became stuck thirteen miles off the highway. The AAA tow truck dispatched to our aid was two-wheel drive and (wisely) refused to come out there. Even some of the street signs are missing. We finally got pulled out by a passing farmer's tractor. I can't speak for the dry roads, but if they are at all wet, give your trip a second thought.