The first thing that needs to be done is figure out power requirements.

Two types of drag occur when an object is moving through water: viscous drag and wave drag. The first is due to the hull "dragging" molecules of water with it, and just like static friction the coarser the hull, the higher the drag. The second is due to energy lost making waves. Hull shape is a big factor, and I'm not doing myself any favors by using a whitewater kayak. This website gives some rough numbers for kayaks so I will go with it. The biggest kayak they calculated has 95 N of total drag at 3.6 m/s. Ignoring all other resistive forces and assuming 100% efficiency, the motor will pull around 350 watts to go around 12 km/h.

Since I want the kayak going about as fast as a bike, I'm going to calculate the drag and power needed at 20 km/h or 5.6 m/s.

This goes to show just how much more power is needed for higher speeds. I'm assuming that the numbers provided by the website include the mass of people in the kayak; otherwise these power estimates will be wayyyyy off.

As a proof of concept I'm using a brushless motor system designed for RC cars that I had lying around. It is rated at around 900W peak. Unfortunately I couldn't find any reliable information on continuous power so we'll just have to try it out.