If it's of any help, we've taken two or three 10 day family holidays in Helsinki, and in June I'd recommend just chilling and taking in the summer atmosphere, which is just great. They really do know how to do summer in Helsinki! It's a young, vibrant city with plenty of places to eat, indoor and outdoor.
In mid-June sunset's not until 2245 and it's great just being outdoors. There are loads of things to do - take your pick, according to how much time you have, and what the weather's like:
https://www.tripadvisor.co.uk/Attractions-g189934-Activities-Helsinki_Uusimaa.html