I picked out a shorter hike from our Derbyshire Walks with Children (#10, Dale Abbey, 1.5 miles). It also was fairly close (15-20 minutes), though not in the Peak District.
The main focus of this hike was to visit The Hermit's Cave. A Derby baker name Cornelius heard a voice (from God?) to lead a life of solitude and prayer in Dale Abbey in the 1100's. So, he ended up in this cave (he actually carved it out, so the story goes). Fortunately, he didn't waste away by himself and eventually built a church, All Saints Church (after a kind donation to get started). An abbey was later built in his honor.
Kedleston Hall. We set out to do the "long walk" (3 miles) but got confused and "only" did the "short walk" (2 miles).
Supposedly the home of Mr. and Mrs. Bert Badger. Fortunately, they weren't around.
One of the many fantastic views out from the Kedleston grounds. The photo doesn't do it justice.
After the walks/picnic we ran into town where we dropped an exorbitant sum to see Kung Fu Panda 2 in 3-D no less. It was quite a day. Alex mentioned many times how it was the best day ever (or at least this week and even with all the walking). When we got home he wanted to see when the next inset day was so we could have another day out together. :-)