If you're constrained by an 11-day visit, I would suggest having a go at Wainwright's Whitsuntide Walk. Over the course of the Whitsuntide break, he came up with a route that would get him to the top of of sight of every lake and summit in the Lake District. It is tough going if you follow Wainwright's distances and timings but, taken at a more relaxed pace, is a lovely way to enjoy the area. In good weather, you can even begin to understand why Wordsworth found the place so appealing.
Alternatively, heading farther north to the Cairngorms is always fantastic, whatever the weather. The combination of mountain, plateau, steep crags, chilly lochs and the Caledonian Forest, along with the freedom to camp and plan your route to suit make the journey to Scotland totally worthwhile. It's the scale, age and emptiness of the area that is the attraction.
