Years ago we used to stay at Scales Farm on the side of Blencathra. It's North Lakes but is probably accessible from Manc in the time allotted since it's very accessible from the M6 (it's on the A66). It's next door to a pub (White Horse Inn) which I think is decent.
No idea on baby friendliness, I'm afraid, though I can't fathom a B&B in that part of the world that won't accommodate families.
EDIT: Just checked their website and they can provide cot and high chair.