Knaresborough is lovely for a day trip, but I wouldn't live there. There are a few rough pubs and empty shop units now and it's only really in the summer that it looks lovely.
Ilkley is pretty close to me. Lovely little town - it has a Lido that we love to go to in the summer.
It also has a train link to Leeds (Ilkely is the end of the line), good shops, park, outdoor lifestyle - walking, cycling, riding, rambling - it has a Betty's tearooms if you like that sort of thing (Fat Rascals...yum!). You are a stones throw from the Yorkshire Dales National Park and Bolton Abbey is just up the road. We are also considering Harrogate for our next move but like you say, you get more for your money in Ilkley and it's still an up-market neighbourhood.
Agree that the drive to Leeds is a pain in the butt - one road and it's slow in rush hour.
Burley in Wharfedale is closer to Leeds than Ilkley but still a lovely village in the same area - we were there yesterday and it's lovely, we took kids out and there were loads of families out and about at the farmer's market, the park and sitting in the beer gardens!
Otley is a bit closer again but it doesn't have a rail link so houses are much cheaper here. It's not as nice as Ilkley or Harrogate though - has a few "scruffy" bits and the town centre is always jammed with cars.
I hope you enjoy your move to here. This is a lovely part of the world to live and bring up a family - I really hope you will be happy here!