You are in for a treat. By all means, include Skyline Drive. The historic value is priceless. The views are breathtaking! A couple of days in the Great Smoky Mountains National Park would be great! Since the end-point of the Skyline Drive is near DC, some time there is worthwhile (Iwo Jima Memorial, Viet Nam Memorial, WWII Memorial, Holocaust Museum, Arlington National Cemetery [especially changing the Guard], Smithsonian Museums). Get a room in Manassas, VA, (nice Battlefield tour nearby) and get transport into DC. Enjoy!
Agreed. Skyline Drive is beautiful.