fitfunc[v_?NumericQ, r_?NumericQ, x_?NumericQ] :=
y /.
FindRoot[((1/L - (x/k)^2*2.16621*10^-31/L)*
   Sqrt[r*y*2*I0/(x*v)]/(BesselJ[0, Sqrt[r*y*2*I0/(x*v)]] -
      BesselJ[2, Sqrt[r*y*2*I0/(x*v)]]))^2 + (v*x*
   Sqrt[r*y*2*I0/(x*v)]/(k*r)/(BesselJ[0, Sqrt[r*y*2*I0/(x*v)]] +
      BesselJ[2, Sqrt[r*y*2*I0/(x*v)]]))^2 == (I0/k)^2, {y, 0.6}];
FindFit[dat, fitfunc[v, r, x], {{v, 0.386*10^-3}, {r, 1.0}}, x]