Thinking about this, there is one western park that hasn't been mentioned. Rocky Mountains NP. It is near the eastern edge of Colorado and might only be a two day drive from KY. Assuming your week includes weekend on both ends, that would give you 9 days. 4 days driving and 5 days in the park. If you haven't been to the mountain west then RMNP will catch your attention and make a big impression on your child.
Boundary Waters is probably also in range but it is a specialized park that requires a boat/canoe to see. That's one we haven't been to yet.