I live here
. 10 days is heaps of time for the top of the South Island. It's about 2.5 hrs from Christchurch to Kaikoura, then maybe 2 from there to Blehneim (and the wineries). What do you like other than wine? Scenery & nature aplenty.
You could go from Blenheim to Nelson (very bendy road), then from there to the Abel Tasman (Kaiteriteri is a nice little town, with Marahau is right next to the national park - you can walk in from there). Nice pizza at the start of the track... There are water taxis that can take you in & drop you off if you want a longer walk - either out of the park or to another taxi stop.
You could go from there over to Westport (little bit of doubling back) and down from there to Greymouth. Punakaiki (Pancake Rocks is in between). There's a great seal colony near Westport. Then south if you want (glaciers), or a small double-back and through Arthur's Pass back to Christchurch. Lake Matheson (the lake with the view of Mt Cook) is nice in good conditions. The forest on the west coast is called rain forest for a reason though
. There are lots of other forest walks near both Fox Glacier & Franz Josef if that's your thing. Nice hot pools in Franz!
Distances are not insignificant & roads nothing like the UK... Queenstown to CHCH without stopping (and you would want to stop) takes about 6 hrs. Arrowtown is lovely - and has a child-friendly miners' settlement that is good fun with kids. Driving from the top of the S Island to QTN would take you quite a long time though!
There is a LOT of scenery... The road from CHCH - Kaikoura is spectacular in places, as are sections of the road on the west coast.
Just N of Kaikoura is an amazing walk into the bush - a stream where baby seals go to play. Loads of them, depending on the time of year. Playing in pools right next to the path you walk on.
DOC (Dept of Conservation) as mentioned above do a great little scheme for kids in national parks Kiwi Rangers but your DD may be too young? We did some when my DD was 5. About really getting touch with nature (taking shoes off & standing in moss; art activities; i-spy and similar).
If you do do the drive from Nelson back to CHCH (rather than going to the West Coast), Lewis Pass is a bit quicker & you could divert to Hanmer Springs...
South of CHCH - Oamaru = steam punk capital of somewhere. Old Victorian quarter. Also there's a penguin colony there where hundreds of penguins appear as if by clockwork in the evening (town has many 'caution penguins crossing' signs...).
Am sure there's more... this is long enough for now though!