My best guess is that we don't have enough information to solve this.
k4 is an angle whose numerical value we know.
k3 is the square of a length whose value we know
k1 is the reciprical of cos k4, so just another numerical value we know
k2 (he's there) is a known length divided by the sinus of a known angle, so just another numerical value we know.
I've looked at the link to the polynomial approach, I can't quite see how to get round the fact that I have an A and the square root of A+constant.