$K(u,v)$ is a simple extension of fields if $u$ is separable
Look at Ken Brown's proof in the separable case: http://www.math.cornell.edu/~kbrown/6310/primitive.pdf
The only place they use separability is to show the gcd of the minimal polynomials has distinct roots. This only really requires one of the polynomials to be separable.
By the way, thanks for this question! I would never have thought this is true.