You're trying to rewrite m in terms of an average density multiplied by a radius, which is OK, but then you need to be careful. If you take your radius to be the schwarzschild radius, the average density is known (in fact, you know the total mass much easier, so this whole procedure is a little silly to begin with). If you try and apply this to an arbitrarily small sphere (say, approaching the singularity), you'll have to say the density goes to infinity, while the radius goes to zero. The product Infinity*0 isn't well defined, so you cannot proceed in such a way.
In reality, the black hole has a well defined total mass, so simply use mv^2/2 (or the relativistic version, if necessary).