We go to Derwent Water every year in Nov and July (timeshare), and our Dd, now 3, has been going since aged 3 mths. Here are some of the things we have done:
Rheged Centre, just off M6, about 20 mins from Keswick, has a play barn area (the only one we have found outside of Penrith). Good cafe, and some interesting shops as well there.
Beatrix Potter exhibition in Bo'ness on Windermere. About 1/2 hr from Keswick. A MUST DO visit for your dd. Excellent exhibition (that sounds stuffy, I mean that all of the Beatrix Potter characters are in settings at little one's height), good cafe for lunch, and if it's quiet, they let you go round twice. Maybe combine with a walk down to the lake, take lots of bread for the swans and ducks.
Keswick Launch - boat trip around the lake. Buy a hop on, hop off ticket. You can get off, and have a gentle walk around to the next boat stop. Ask advice at the ticket office on which stops to get off at.
Aquarium of the Lakes, Newby Bridge.(Bottom end of Windermere). Good fun, lots to look at. Combine it with a train trip, which stops at the Aquarium.
Priests House, Caldebeck. Fantastic scenic drive high up into the mountains, for lunch. The Priests House is great food, lovely setting and a great village to stroll around.
Bassensthwaite lake is a gentle walk, as is Buttermere.
I believe that there is a good pool in Penrith - never used it. There is also the Oasis Park in Penrith (I think it may now be part of Centre Parcs)
Keswick itself is quite nice for a stroll around - some nice tea shops, and Friars sweetshop is good.
Ambleside is also good to stroll around. Lucy's on a Plate is good to eat, and welcome children.
The Horse and Farrier at Threlkeld is a great place to eat (rustic pub). It gets really booked up though, so book before you go. About 10 mins from Keswick on A66.
Carlisle is good for shopping, but a bit of a trek. We visited Whitehaven (on the coast) in July, but were disappointed. The tall ships were there, but even with this, we found it hard to find anything to do.
If you want an amazing drive for scenery, try Honister Pass. There is also a slate mine worth a visit on the pass, it's very high. Lots of little tea shops in villages on route.
Is this enough! Enjoy you holiday, we love it, and look forward to our next 80 years of visiting. We never get tired of it.