Start small. On the first evening nobody wants a mountain, so walk up to Cogra Moss. It is about fifteen minutes from the park, a small reservoir sitting in a bowl under Blake Fell with a flat track all the way round and a lot of trout in it. On a weekday you will very likely have it to yourself. This is also the walk to do if the forecast is bad and you want to be back inside within the hour.
Day two, go and look at a lake properly. Loweswater is fifteen minutes and is four flat miles round, through Holme Wood, and it is the one lake in the district whose water flows inwards toward the mountains rather than out. It takes a couple of hours at a dawdle and is fine with a pushchair on the eastern side. Finish at the Kirkstile Inn, which sits under Melbreak and brews Loweswater Gold. Book a table if it is a weekend.
If you only drive one road while you are here, make it the one over Fangs Brow into the Loweswater valley at about seven in the evening.
Somewhere in the middle of the week you will want Buttermere, because everyone does and because it deserves it. Twenty five minutes over the tops. The circuit is four and a half miles, almost entirely flat, and includes a short tunnel a Victorian landowner had cut through rock so his guests got a better view. Go before ten or after four, and if the main car parks are full, park at Lanthwaite and walk in along Crummock.
For a real day out, Ennerdale is fifteen minutes and has no through road, which is exactly why it stays empty when everywhere else fills up. The lake shore path is stony rather than groomed, so proper boots. Pillar and Steeple sit at the far end if somebody in the party wants a mountain.
And when it is properly hammering down, which it will be at some point, drive to Whinlatter. It is the only true mountain forest in England, the trails work in rain, there is a cafe and a decent visitor centre, and Go Ape is up there if the children need wearing out. Failing that, the Beacon Museum on Whitehaven harbour is a solid two hours, and Wordsworth House in Cockermouth is small, cheap and better than it sounds.