I booked a cottage through Lake Lovers and it was very good - have a look at their website. All the properties seem pretty high quality.
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https://www.lakelovers.co.uk/url]
The kids are a bit young for hiking but there are some nice gentle walks round various lakes that they might enjoy, with opportunities for paddling, throwing stones into the water etc. You can also go on boat trips (either rowing or big steamers but it might be a bit chilly). There's really only one main road through the National Park, unless you detour via the M6 so you need to choose where you want to stay carefully. If you choose the Southern Lakes then the Northern ones are a bit of a trek and vice versa. We stayed in Grasmere which is beautiful and very well placed as it's pretty much in the middle so it makes it easier to get everywhere. It also has a particularly nice gingerbread shop!
Having said that I'm not sure I'd have taken my kids there when they were very small. It's such an outdoorsy place, and the scenery is so beautiful, but that's really lost on young children. North Wales might be a better bet - still beautiful scenery but with the added bonus of castles to explore, steam trains to ride on, a mine to go down, Portmeirion (recreated Italian village) - I just think there will be more to do there than the Lakes for the little ones. I took my kids there when they were about 3 and 5 and we had a great time.