We go to the lakes a lot, sometimes self catering, sometimes hotel with our 1 year old and 3 year old.
Keswick is an absolute favourite, lovely walks, coffee shops, shops, pubs and more low key than bowness, grasmere, Ambleside etc. Lovely that you can walk from town to the lake, and there is a really easy walking path round the side of the lake.
Coniston is also fab, we stayed here for a week with a very young dd2 and it was AMAZING. There is a gorgeous bathroom, amazing views, super comfy and a pool room which was lots of fun when the kids went to bed. www.lakelovers.co.uk/property/the-farmhouse/
For shorter breaks of 2/3 nights our absolute favourite is The Swan at Newby Bridge. Super child friendly with a pool, walking trails and great food, and a fab steam train literally just down the road. Staff are all lovely. They have brilliant family rooms with either a bunk room off the main adult bedroom or lovely 2nd bedroom with dolls house, PlayStation and Beatrix potter books. The adult rooms and bathrooms are large and are gorgeous, you don't feel you're second best cos you're in a family room. And they bring the kids free milk and cookies at bedtime which is very sweet! www.swanhotel.com
For a day out, visit Allan Bank in Grasmere, the least stuffy, most child friendly NT place ever- they have a lovely huge art room which is all free, and a great kids playroom too. Grasmere is a great day trip in general.
Hope that helps, enjoy your break.