Please edit your code. Both time and sdata are undefined. Also, you mention that there are "plotting errors" but show no errors nor is there a description as to what the errors are.
The model and the data can only estimate R/J and not R and J separately. Evidence of this issue (beside observing the structure of the model) is that the estimate of the parameter correlation matrix is singular:
nlm = NonlinearModelFit[data, {k[t] /. \[Omega] -> 350.73,
{0.0001 <= R <= 0.0005, 0.0013 <= J <= 0.0018}}, {{R, 0.0003}, {J, 0.0015}}, t]
nlm["BestFitParameters"]
(* {R -> 0.000127627, J -> 0.00162485} *)
nlm["CorrelationMatrix"]
(* {{1., -1.}, {-1., 1.}} *)
