The conversion cannot be exact, because, for arbitrary u and au in W, the expression ConvertThis is not necessarily constant on each element, while K is the space of elementwise-constant functions. Thus, there must be some approximation, the choice of which is not unique. A natural choice when K is a degree-0 DG space might be to approximate ConvertThis by its average on each element, which would be equivalent to the L^2 projection operation implemented by
ConvertThis_K = project(ConvertThis,K)
Note that project involves a linear solve, but, when projecting onto DG spaces, it can be optimized by using independent solves on each element, as demonstrated here.