I believe that you must mix in some time in the countryside. I suggest 12 days, splitting them between Vienna, Salzburg and Prague or Vienna, Graz and Budapest.
Vienna deserves 2-3 full days. Salzburg and/or Graz deserve 2-3 full days. Budapest and/or Prague deserve 2-3 full days.
It is best to travel by train between these locations, both for cost and for your overall sanity. I would suggest renting a car in Salzburg and/or Graz to make the best use of your time.
In Vienna, I would suggest visiting Schonbrunn (Labyrinth and Zoo), Technical Museum, Haus Der Musik, the Rathaus film festival for dinner, climb St. Stephen's, ride the Ferris Wheel/visit Prater, visit Kahlenberg and walk down the hill to a heuriger, Natural History Museum, Vienna Museum. Eat lots of cake and sausages. I also have kids and these are the things they like to do.
In Salzburg, as noted above, I would suggest renting a car for visiting Hallstatt and the Salt Mines as well as Wolfgangee. Be sure to visit the fortress in Salzburg and Hellbrunn gardens. If you decide to visit Graz instead, stay in the Loipersdorf area which has a terrific water park, but most importantly, the best attraction in all of Austria - the Zotter chocolate factory. In addition, Riegersburg fortress is a fantastic visit and also right there.
Make the trip about making memories, not about checking things off of a list that you are "supposed" to see.