One of the comments I tend to get in summer is that, because I live in the mountains, the temperatures must be cool. Granted, it may be cooler than the cities but it is still hot and humid. Help is at hand though with the swimming pools and water slides at Leisure Pool Aurora.
It is a small, town-run leisure complex so probably not worth traveling vast distances to experience, but if you are in the area it is a great day out that will drain some energy out of the kids. There are two water slides (a long, fast blue one outside, and a twistier yellow one that spits you out into a pool inside), and three pools both inside and out. It is very child-friendly with shallow areas for paddling and plenty of places to kick back and relax. The main building has a cafeteria too so you can quite easily camp out for the day. It is a popular spot in summer but the complex is large so it doesn't feel crowded.
Leisure Pool Aurora is part of Chuo Park which has an abundance of tennis courts, a couple of baseball diamonds and a running track , so if you need space to throw/kick a ball around you are spoiled for choice. There is also a free foot onsen (though the attraction of bathing your feet in hot water while it is thirty degrees in the shade is lost on me.) The park is a the bottom of Iwappara Ski Resort (and in the winter, when the whole area is covered in snow, it becomes a snow mobile course which starts in the Aurora car park) and has rivers running through it and lots of green all around.
If you are looking for other options to cool off there is a popular BBQ site just down the road where the kids can paddle in a stream, or if you want somewhere deeper you can join the locals close to Iwappara Skijomae Station where there is a great swim-hole in the river and rocks to dive off.
Aurora water slides open mid July each year and the inside pools are usually open from the start of June. If the muggy summer weather gets to you, put it on the list.
