You're out in the sticks, so plenty of walks, but well within an hour's drive for Salisbury, Bournemouth, Christchurch etc.
The New Forest is not far out to the east, just keep going when you get to Fordingbridge. Burley would be the closest main village.
Brock, Beaulieu and Lymo further S/SE. Not sure when the Pigs get released this year, but they're normally up North, around Brook.
Monkey World down at Wool would also be good for your daughter. Then the Tank Museum is not far from there. Then S and SE, for
Durdle Door, Corfe Castle and Swanage (and the Steam Train in between). The chain ferry to Poole is still down, at least it was a couple of weeks ago when we were there, but Studland on the way to it is a lovely beach if the weather's nice.
There's Stourhead to your NW, but it's too early for leaf kicking and the changing colours. Pricey too, as you now have
to pay for the dull house visit, as well these days.
There's a tap room at Sixpenny Handley brewery, which is close to you, if that's your thing and Ringwood brewery, if you're really keen.