You're right I ought to have add the initial x-velocity to obtain the final x-velocity. Then determine the derivative (although as put it was irrelevant)In your equation for vx, you left out the initial velocity of the skater. Otherwise, your first two equations are correct. Your second two equations (taking derivatives of the velocities) were not done correctly, and were not even necessary for solving this problem. Once you make the correction to the equation for vx, take the sum of the squares of the x and y components of velocity and set them equal to the magnitude of the final velocity you are seeking. This will give you an equation for solving for the angle θ (or sinθ or cosθ).
Let me work on your question.