Seattle and kids can be a good mix.  A total pro-move that most won't consider is doing a factory tour at Boeing up in Everett: http://www.boeing.com/boeing/commercial/tours/index.page

The Museum of Flight East Lot is also a real kid-pleaser, as is the Seattle Aquarium, which is on the water downtown.  The Pacific Science Center has a cool outdoor area (if it's hot, have the kids wear swimwear and they can splash around outside).  And the Experience Music Project is fun-- crazy weird looking on the outside and full of geekery (music and sci fi!) on the inside.

Transportation-wise, there's a decent bus system and a silly monorail that only goes a few miles but is actually really handy for the Sheraton if you are heading towards the EMP, Space Needle, and Pacific Science Center.  The Center for Wooden Boats is a nice area and has the new Museum of History and Industry right there on the water.

You can also take the kids out on the Elliot Bay-- either the Water Taxi Route & Harbor Ave SW ($4.75/person to West Seattle, which can be fun-- has a shuttle to Alki Beach Park), Let's Go Sailing, or Argosy Cruises are good bets.

Beth's Cafe will give your kids paper, crayons, and a spot on the wall for anything they draw during breakfast - the walls are covered. After you finish your 12-egg omelette, take a walk around Green Lake Park. There is a wading pool for your kids, and you'll find lots of little chocolate shops, ice cream parlors and cafes around the lake. By bus, it's around 30 minute north of Seattle. On your way back, stop by the Fremont Troll.

Make sure you visit the Pike Place Market, and pass by the seafood stand near Rachel the Piggy Bank. There's always an ugly looking fish in the middle of the ice - you'll know it when you see it - and you should tell your kids to check it out. Keep your camera ready for the reaction-shot. 

From the Market, walk down to the Seattle Gum Wall and make a contribution. Then check out the otters at the Seattle Aquarium and catch the King County Water Taxi to Alki Beach Park.

Don't forget to visit the Seattle Center. Your kids will love the Space Needle, and if it's a hot day then they can cool off in the giant water fountain. If they aren't soaked, check out a play at Seattle Children's Theatre or spend a couple hours at the Pacific Science Center. You'll probably enjoy the Experience Music Project even more than your sons, and the pickup point for Ride the Ducks of Seattle is only a few blocks away.

Have fun!

Take public transportation from the airport to Sheraton Seattle Hotel. You are staying in a great area! One morning walk to Bainbridge Island Ferry and take it to Bainbridge Island .. 2 Blocks from the Bainbridge Ferry Port is Streamliner Diner..cool old time diner and GREAT Breakfast for all ages..you can time it to finish meal and make it back to the next ferry if you want to rush your meal.  Walk along the Sound along Pike Place Market (Original Starbucks there) and watch the Sea-Planes Land, Art Work or rent a bike to ride along the Pier North. Space Needle, Experience Music Project all walking distance from your hotel...Feeling Adventurous? Rent a car from Hertz Downtown and drive to Mount Rainier National Park hike for few hours and drive home..about 2 hour drive. Can also go see where Boeing Planes are assembled at Boeing Commercial Airplanes in Everett. If bored Uber Ride to REI huge store and Climb the wall or have kids do it! 

Try the Museum of Flight - kids will love it.

Just wanted to add the Living Computer Museum, I hear it's cool but haven't been yet. :-)

Great answers here. One more thing to think about is the Columbia Center Sky View Observatory. It's the tallest building in Seattle and when you're up there, you look across the city and down on the Space Needle

The Columbia Center is the black one on the right, well above the space needle. 

Ditto to everything Christian and Tony recommended!  Here's my take on a few of these:

• Definitely Pike Place Market.  Take a picture of the cute gold/bronze'ish pig statue in front, then walk right past it till you hit the wall and take a flight of stairs down to your left where you'll hit the Seattle Gum Wall.  Let your kids leave their ABC gum there.Once you get back to civilization upstairs, watch the fishmongers toss fish
  Pay a visit to Golden Age Collectibles Ltd to look at comic books and toys.Get some made to order mini donuts at Daily Dozen Donut Co.  So much fun to watch the machine go, and you get to pick the toppings!
• Your kids will love a ride up the Space Needle.  The outside is fun to hang out at, too.
• The Ride the Ducks of Seattle is fun.
• If you have time, another fun water activity could be riding the ferry over to Bainbridge Island.  Your kids could run around somewhere nature-y.  This would be an afternoon or day trip.
• A couple really cool museums are the Museum of Flight and the Experience Music Project.  The outside of the EMP would be fun for the kids to run around.

Have a blast!