Drink and or eat at the Airport. That's what I did there with a several hour layover when I went to Alaska. The airport itself is a bit far from the city that I wouldn't risk going into the city for anything less than 7 or 8 hours.
Denver airport is kinda deep from the downtown area. And even then, it’s just a lot of places to eat and drink, because the city itself isn’t nestled in any mountainous hiking areas or anything like that.
Apparently there's a kids play area at DEN. I'd plan bathroom, food, play area, bathroom, airport subway ride/people mover/other airport stuff kids like and there isn't always time to play on, snack, bathroom, chill at the gate with a tablet... in no particular order.
Denver Airport Train & Light Rail | VISIT DENVER