Cannot fit data due to NIntegrate errors

Question

Here is the data I am trying to fit

``` data={{0.06, 1.01639, 0.0121617}, {0.13, 1.00026, 0.00916985}, {0.2, 
      0.908723, 0.00828259}, {0.275, 1.01714, 0.00821439}, {0.35, 1.03227,
       0.0143054}, {0.5, 0.959998, 0.0157588}, {0.75, 0.930953, 
      0.0135328}, {0.875, 0.837181, 0.00844368}, {1, 0.788837, 
      0.0117245}, {1.125, 0.731066, 0.00903227}, {1.175, 0.710571, 
      0.0122025}, {1.25, 0.529901, 0.016269}}```

The first is a temperature, the second the superfluid density, the third the error.

Some manipulations to get the data and the error set up for the fit:

    ```nn = Drop[data, None, {3}];
        m = Length[nn];
       en = Table[{data[[i, 1]], Around[data[[i, 2]], data[[i, 3]]]}, {i, 1, m}];
     errn = Flatten[Take[data, All, {3}]]```

Some parameters:

``` \[Xi] = 116;
\[Lambda] = 5952;
\[Kappa]  = \[Lambda]/\[Xi]
       Tc = 1.85;```

The functions I need for integration:

```En[eps_, \[Phi]_, T_, \[CapitalDelta]0_] := Sqrt[eps^2 + Abs[dk[\[Phi], T, \[CapitalDelta]0]]^2]
dk[\[Phi]_, T_, \[CapitalDelta]0_] := \[CapitalDelta]0 Sqrt[2]Cos[2 \[Phi]]Tanh[1.74 Sqrt[1 - T/Tc]]
\[Rho]s[T_?NumberQ, \[CapitalDelta]0_?NumberQ] :=1 - 1/( 4*Pi*T)NIntegrate[(1/2(Cosh[2 En[eps, phi, T, \[CapitalDelta]0]/(2 T)]) + 1)^-1, {eps,0, 100 T}, {phi, 0, 2*Pi}]
Ts[\[CapitalDelta]0_] := \[CapitalDelta]0/\[Kappa]
ns[T_, Ts_, \[CapitalDelta]0_]:= 1 - (1 - \[Rho]s[T, \[CapitalDelta]0 ])*((Tc + Ts[\[CapitalDelta]0])/Tc) (T/(T + Ts[\[CapitalDelta]0]))```

When I plot the function ns I am starting to get an error:

```Show[ListPlot[Table[{T, ns[T, Ts, 2 Tc ]}, {T, 0.01*Tc, 0.95*Tc, 0.05*Tc}],Joined -> True], ListPlot[en]]```

However,it still shows the function.

But if I try a Levenberg-Marquart fit, it fails:

```one = NonlinearModelFit[nn, 
ns[T, Ts[\[CapitalDelta]0], \[CapitalDelta]0], {{\[CapitalDelta]0, 2 Tc}}, T, Weights -> errn, Method -> LevenbergMarquardt]```

The integral is 1 over the square of Cosh, which is a well behaved function, so I am not sure, where I am going wrong.

Thanks a lot for helping.