We have a pretty robust baby who will be around 18 months in spring.. and my good lady doesn't do much cycling due to asthma so they will want other distractions on the days that I will be on the bike..
I also havre asthma as do a lot of other riders/sports people. Take it is not controlled or v-bad?
You could try the YHAs at Ambleside or Keswick for family rooms.
Keswick and Kendal have swimming pools.
Lots of Lakes with boats/steamers on them.
Staveley has Wheelbase bike shop, Wilfs cafe, a brewery, a deli with some lovely cheeses, little bakeries
Kendal has Evans bike shop
Ambleside has Biketreks
Keswick has Keswick Mountain Bikes.
My wife does not bike due to other med probs but we can always find plenty to do. Taking grandkids to the wildlife centre across the Lake from Windermere.
Grizdale has a new visitor centre, play area and bike shop.
There is a narrow gauge railway, forget where.
A slate mine
Touristy open top bus
Plenty to keep everyone occupied.