I am a physicist at the UK's National Physical Laboratory and I am involved in the redefinition of another unit - the unit of temperature. The work concerning the redefinition of the kilogram is >much more philosophically fraught and technically difficult. I understand many of the rather skeptical comments expressed here
The current situation is unsatisfactory because the mass of the kilogram is changing, albeit by a small amount - a few micrograms in 1 kg - i.e. a few parts in 10^9. The aim of the work is to replace this artefact with (essentially) a procedure.
Effort onne (using the silicon sphere) is essentially trying to build a link between a macroscopic mass, and the microscopic masses which we expect to be fixed.
The other effort not mentioned here is called the watt balance which is a machine which can exhibit the same inertial mass as a kilogram - or any other weight.
At the moment the two watt balances disagree with each other and they both disagree with the silicon sphere result. There is still a lot of work to do on both approaches.
Incidentally, the reason is it is a sphere rather than a cube is because of edges. Edges are amazingly fragile, hence the sphere.
ALl the best