I think the only time the speed of the car should matter is if it is going so fast that it skids. Otherwise, static friction between the road and the tires will be maintained, which means it will follow the same path regardless of speed.
The front tires will only turn on one axis (per tire). As such, the only way the car can move without skidding is for the tire to move in exactly the direction it is pointed, regardless of speed.
The angle of each of the front tires is carefully calculated so that all four tires are tangent to a circle about the same center point at all times. From that information, you can calculate the turn radius of the car with geometry. Which appears to be the approach you've taken.
It would probably be more useful to work with the radius of the circle in which the point directly between (and equidistant from) the two rear wheels moves rather than the point directly between (and equidistant from) the two front wheels. This allows you to work with the dimensions of the car (which will be given) instead of the angle from a point on the car to the center of the turn (which will not be given). This also has the advantage of making your triangles right triangles.