I can't imagine tackling art museums after an overnight flight, and I love art. If you're thinking about some kind of tourist card that includes a sightseeing component, I would discourage that. I'd get transportation into the city and plan to use the jetlag-fighting technique of walking around outdoors. Rick has a walking tour laid out in his book. The tourist office offers guided walking tours, but I'm not sure what time they depart. Your kids may be a bit too young for those, especially after a night on the plane.
A trip to the San Miguel Market near the historic center may be helpful; you'll be on a weird eating schedule after the flight. San Miguel is a touristy sort of place, not a cheap option, but it may be what you need on arrival day. I don't remember a churro stand inside the market.
Retiro Park is large and free. It's near Atocha Station. There's a nice botanical garden nearby, but there's a fee to enter that. That's where I spent a couple of hours on my arrival day in 2016. But with young children, I think Retiro would probably be better since they'd be able to run around.
You'll need a place to stash carry-on bags, yes? One option is Atocha Station. I have not used the luggage-storage facilities there and cannot tell you how long drop-off and pick-up will take. If you decide to head there, a train or the airport bus may be more convenient than dealing with the subway from the airport. I assume there won't be taxi-vans at the airport that would accommodate 7 people, else that would probably be an economical choice for your group.
