Papa:
We did the trip last summer and were on the road from Sunday, June 24th through Saturday June 30th.
Sunday: We traveled 227 miles from our home east of Atlanta up to Cherokee, NC and accessed the Blue Ridge at the start. We overnighted at the Comfort Inn in Black Mountain NC. Average speed is really low due to Sunday drivers, a lot of tunnels and many overlooks to stop at. My guess is we averaged under 30 mph as we were on the road for eight hours.
Monday: We got back on the Parkway at mile marker 360 and covered 215 miles to Floyd, Virginia. We stayed at the Hotel Floyd in the downtown area and can highly recommend it as a place to stay. Unfortunately the two good restaurants in Floyd close on Mondays so we ate at the Blue Ridge Cafe that we cannot recommend. There is a Hardies in town. (Note: do not trust your GPS to find a quick back road off the Parkway to Floyd).
Tuesday: Back on the Parkway at mile marker 160. Lots of things to visit and sight see on this leg. We got off the Parkway where it ended at Waynesboro and took highway 340 to Luray, Virginia where we stayed in the Days Inn. We toured the caves and took a few side trips off the Parkway. We logged 221 miles that day.
Wednesday: Up early and took Highway 211 into the Shenandoah National Park at mile marker 33 and then traveled the Skyline Drive south through that part of the park back to the Blue Ridge. We traveled 175 miles that day and stayed at the Peaks of Otter Lodge run by the Park Service right on the Parkway at Mile Marker 86. Reservations are needed there and gasoline is scarce. The place is old, quiet has a great restaurant, and some good wines.
From there we stayed on the Blue Ridge for two more days staying at the Skyline Inn and Tavern in Spruce Pine, NC (Built into the side of a mountain in 1948 at Mile Marker 331) on Thursday and at the Brasstown Valley Lodge in Young Harris, Ga on Friday, coming home on Saturday.
One of the things that hurts average speed is having to get off the Parkway to find food and gas. There are no convenient places so sometimes it's 25 or 30 miles round trip to get off and look after these things.
If I were you I would plan to get off the Parkway at Lexington, Virginia and take highway 64 west through the mountains of Virginia/West Virginia heading back to Chicago.