Just to get you going.
Let the curve be . Suppose (without loss of generality) c is already parametrized by arc length. Then the formula gives , so there exists a function such that . To explicitly calculate this function, recall that .
Let k(t), construct the curve a(t), for which the curvature at is equal to k(t) and which satisfies the initial conditions a(0) = (0,0) and a'(0) = (1,0).
Now, if I substitute 0 in for t, I get (0,0). But taking the prime, should I take it relative to t or s?