Jacobian matrix normalization
You can't.
The Jacobian matrix performs the change of variables for you. If your starting variables have different units from the resulting variables, then your Jacobian matrix must not be dimensionless.
For example, simply consider the difference between polar and cartesian coordinates in the plane. The cartesian coordinates measure two lengths. The polar coordinates measure an angle and a length. Were the Jacobian matrix dimensionless, you would have that angles and lengths are measured by the same unit, which on a euclidean geometry is absurd.