Agree with ssk44, except for the part about going below the dam.
There is a serious problem with doing a "dam-down" to Lees Ferry in Glen Canyon. The problem I see is that you want to take a small child onto a river where the child will likely die of hypothermia within a few minutes after falling into the extremely cold water. This is not a place for small children at all. It is just too cold. I can't emphasize that enough. If you capsize, your kid could die pretty quickly. Doesn't matter if the air temperature is 110. The water is 45 degrees, and that doesn't change. I have seen people get into trouble on this stretch of river, and it wasn't pretty. There are a couple of companies now which provide the "backhaul" services. One is Wilderness, the other is based at Vermilion Cliffs Lodge. But that's if you leave the kid at home.
I would say you could go car camp at Lone Rock, if you have a 4wd. It is VERY crowded, but there are lots of kids playing in the water there, and you can kayak nearby.
But long multi-day kayak trips on Lake Powell are exactly as Ssk44 describes: You need an actual sea kayak, 14-19 feet long, with a rudder or skeg, and a lot of paddling skill and knowledge. It can get very rough, very quickly, and there is often nowhere to land, with vertical cliffs going into the water. You didn't mention what type of kayak you have.
I would instead recommend lower Lake Havasu or Lake Mohave. Same problems with wind, but lots more places to land, lots of beaches to camp on, and once you are on a beach, it's yours, and other people will go somewhere else. Lake Havasu has over 100 developed boat-in campsites along its Arizona shore. All are south of the city, all the way to the dam. You can launch in the city at Rotary Park, paddle south, and very soon you will start seeing campsites with picnic tables. (You are supposed to bring cash, put in the "iron ranger.") Lake Mohave has undeveloped camping which is legal on almost 200 beaches, and many, many places to land if it gets too windy. For that trip you can launch at either Katherine Landing or Princess Cove.
Lake Mead has similar problems as Lake Powell. Lots of cliffs, fewer places to land in an emergency.
All the lakes have many speedboats, lots of choppy water, lots of wind. But at least on Mohave and Havasu you can find a place to land. The lake waters are warm enough to swim in all summer long, and into the fall. Plus, they don't have that ugly bathtub ring like Lake Powell has. When it's very busy with speedboats, I just follow the shoreline. This has proven to be a very safe way to go in Lake Mohave and Lake Havasu.
A safer river-running option would be Black Canyon, below Hoover Dam. You need an outfitter to launch you, (Desert Adventures, or several others) or paddle upstream from Willow Beach. The water below Hoover Dam is chilly, but not nearly as cold as Glen Canyon. Although it is very popular, there are a few beaches where people don't usually camp, such as Crane's Nest Wash. The reason is that most people go for the hot springs. If you camp on a beach with no hot springs near it, you will have the beach to yourself.
Maps of all the lakes and the Lower Colorado River are available from FishnMap company. Other info: Lake Havasu camping, call BLM Lake Havasu office. Lake Mohave is administered by the Lake Mead National Recreation Area, so you can call them for info.