I have never used a kite, but have used pop up sails. I suspect with a kite you could only go downwind, as you don't have any way for the boat to bite into the water to keep from just being pulled sideways (and the pull could possibly flip you pretty easily).
For downwind, I'd probably stick to something like the Wind paddle sails (I have one) or the one from Advanced Elements.
If you are looking to sail cross wind, then you may want to look at one of the more advanced sales (Flat Earth, etc.). Some of these come with drop down dagger boards. I've not used any of these, except at demo days.
A kayak with a rudder is much easier than one without. I have used the Wind Paddle on boats without, but it is a bit of work to stern rudder with your paddle while holding the sail up.
Kayak & Canoe Outriggers