The expression model[cd, kvc][[k]][[0]][t] can be simplified as model[cd, kvc][[k]]. The factor dl cancels out, doesn't it? I am not expert in model fitting, but the dependence of your model in cd and kvc seems too involuted. I would suggest incorporating the two parameters into the model using ParametricNDSolve, of which, sadly, I know very little.

Ah I see thank you that works. As a quick follow up question, my next line of code is this, and I seem to get some new errors. Would you mind taking a look at it? (I am assuming you do not need dampdata to debug the code?)

nlm = NonlinearModelFit[dampdata, 
  ArcTan[(dl Sum[Sin[model[cd, kvc][[k]][[0]][t]], {k, NN}])/(
   dl Sum[Cos[model[cd, kvc][[k]][[0]][t]], {k, NN}])], {cd, kvc}, t, 
  Method -> "Gradient"]