I’m not sure I’m following the question, but I’ll try to answer.
We can measure the value a, defined in the slides. This a depends on mu, B_J, I and J. We considered B_J as being known, or something we can calculate. I assume this is the same for the new isotope (same for J), and hence also known from earlier. We are given the value of I, and hence from our measurements of a, we can determine the value of mu.
I’m not sure about this, I was also thinking we could measure the energy splitting, maybe this would give us also some information we could use, but I don’t see how exactly.