Well in August we can safely exclude all southern countries cause chances are you won't be able to enjoy anything in the heat. Don't want to risk it. Scandinavia is fun but steep on a budget - so not a priority on the list. I get the feel from your post you don't really want to be in a big city (correct me if i'm wrong) so no Paris, London or Berlin.

Amsterdam is a good bet - cheap flights, kid friendly, not too hot in August. Plenty to visit and do - but research some day trips, I feel 1 week might be a little bit too much (unless you're really taking it easy). It's a relatively small city so you'll do the main tourist things fast IMO.

I find Copenhagen has a similar feel as Amsterdam, but less overwhelming tourist-wise with still plenty to visit. Amsterdam can get a little packed with stag parties and stuff sometimes, but that's just my impression. Brussels also fits the criteria. If you're feeling adventurous, I really liked Riga in Latvia. Architecture is amazing, it's a little different from everywhere else, I really enjoyed the food + there is a beach not far.