Hello from the edge of the Lake District (Kendal to be precise).
I wouldn't have thought you want to be out in the sticks either, not with a 2 year old. But, if you did want to be somewhere remote, there are endless opportunities as the first respondent suggests. The outskirts of the 'honeypot' villages of Windermere, Bowness, Ambleside, Coniston, Hawkshead, Grasmere and Keswick are all places you could look.
We have a holiday place also that you might like to look at but it certainly wouldn't qualify as remote: http://www.kendalcottages.com - it's a town centre location so near all amenities but peaceful, too, because it's on the river. Possibly not what you're looking for, but Kendal does have a soft play centre, a cinema (and other events) at the Brewery Arts Centre, and the World of Beatrix Potter Attraction / South Lakes Wild Animal Park / Lakes Aquarium / etc. are all within easy reach and great for little ones for those wet weather days.
Kendal also has an Evans Cycles and Askew Cycles hire bikes at a competitive rate. Plus Wheelbase mentioned above is just up the road.
Whatever you decide on, I hope you enjoy your visit to this beautiful county. 
