The Lake District should need no introduction as a paradise for walkers. There is a set of Wainwright’s guides in the cottage that you are welcome to use (although please do keep them dry!) and some maps. If you are not used to planning your own walks, there are numerous guide books available with walks of various lengths. There is some general information here from the National Park, including some guided walks, and some ideas for walks here. And stay safe on the hills – there is a lot of good safety advice here, from the folks who don’t want to have to come and rescue you!
There are plenty of opportunities for both road rides and mountain biking. If you bring your own bikes there is a lockable outhouse to keep them in, a track pump, and an outside tap if they need washing down. There is some information here about cycling in the lakes, including hire locations. Bikes can also be hired in Windermere, Ambleside, Grizedale, Coniston, and other places. (I haven’t hired bikes in the Lakes, so I can’t recommend anywhere.) There is a large bike shop in Staveley, one in Ambleside and a couple in Kendal.
Limestone Cottage has its own Strava identity with some suggested routes. Use email@example.com to log in, with the password CyclingRocks, then go to Dashboard and then My Routes for some road route ideas.
This National Park page has advice about boating and water safety, and links to places to hire boats.
There is a Go Ape! at Grizedale for tree-top adventures.
Experienced climbers could visit the Kendal Climbing Wall. If you have no experience contact them first to find out about training sessions, as you cannot just turn up and have a go.
Various people provide adventures such as gorge walking, or can teach you to navigate or organise other days out. This one I know personally and can recommend – some others are here, and here. No doubt Google can turn up a few more!