As I understand it the problem was not so much to find a mathematical definition of the kg based on other units but rather to find a definition suitable for experiments. If you can't practically use the definition to reproduce the prototype then it's not a very good definition indeed.
It seems that they settled on this definition because a Kibble balance[1] has shown to be precise and practical enough:
> Accuracy criteria were agreed upon in 2013 by the General Conference on Weights and Measures (CGPM) for replacing the current definition of the kilogram [...] with one based on the use of a Kibble balance. These criteria have since been met, and the definition of the kilogram and several other units will change on May 20, 2019, World Metrology Day, which celebrates the establishment of the SI, or metric system, in 1875, following the final vote by the CGPM on November 16, 2018.
It seems that they settled on this definition because a Kibble balance[1] has shown to be precise and practical enough:
> Accuracy criteria were agreed upon in 2013 by the General Conference on Weights and Measures (CGPM) for replacing the current definition of the kilogram [...] with one based on the use of a Kibble balance. These criteria have since been met, and the definition of the kilogram and several other units will change on May 20, 2019, World Metrology Day, which celebrates the establishment of the SI, or metric system, in 1875, following the final vote by the CGPM on November 16, 2018.
[1] https://en.wikipedia.org/wiki/Kibble_balance