In the simplest variant, from the slope before cmc one can get the specific conductivity due to the monomer. From the slope after cmc, one then obtains the conductivity due to the micelles. The later is related to the charge and the radius of the micelle through Einstein-Stokes equation. The Einstein-Stokes equation is too approximated to use for highly charged micelles, though. Some variant of Onsager-Debye conductivity model is perhaps a better option.
Also I - once you separate the conductivity due to micelles, by comparison with the theoretical Einstein-Stokes conductivity you'll get (quite inaccurately) size, number of micelles, and aggregation number.
For aggregation number, I would recommend you to use the pyrene fluorescence measurement method using a quencher (You may use cetyl pyridinium chloride)