This is one of the most beautiful areas in North America, and I agree you would enjoy more than nine days. However, your personal preferences will determine how you split the time among the various parks. Our last visit, our fourth trip there, we spent four nights in Jasper and eight nights in Banff, visiting Lake Louise, Moraine Lake, Yoho NP and Kootenay Np as day trips from Banff. Also, we spent time in Kananaskis Country (great scenery and wildlife, few people). Three and six might be a good split for you.
The Icefields Parkway (between the parks) is a gorgeous drive with lakes, glaciers, waterfalls and wildlife all along the way. Some of them deserve an early morning visit, which means separate visits, as opposed to a simple stop as you drive the parkway.
Jasper has numerous lakes, a beautiful downtown, a great lodge, Maligne Canyon, wonderful hiking and lots of wildlife on almost any road you drive (so drive all of them). If your kids swim, make sure they get to Horseshoe Lake. Edith Cavell Glacier is another must see. They will enjoy the gondola ride to the top of Whistler Mountain.
Banff is a great town, but often very crowded in summer. Drive the Bow River Parkway to visit Johnston Canyon (several waterfalls), Lake Louise and Moraine Lake. Peyto Lake, Lake Herbert, Bow Lake, Waterfowl Lake and Vermilion Lakes (right in Banff) are all good morning spots. Vermilion Lakes is also where sunset is usually good. Drive the loop that goes by Lake Winnewanka, and several others. Visit Yoho and Kootenay if you have time and feel like another drive.
Check my pbase gallery for shots of most places. Link is below, scroll down to "Canada", then select Alberta. Let me know if you have specific questions.