For[k = 5, k <= 7, k = k + 0.1,
  sol =
    ParametricNDSolve[
      {Y''[x] + k^2 (1 - 0.1  Conjugate[Y[x]] Y[x]) Y[x] == 0,
       Y[1] == N E^(I k), Y'[1] == I k N E^(I k)},
      Y[x], {x, 0, 1}, {N},
      AccuracyGoal -> Infinity, MaxSteps -> Infinity];
  Ns = FindRoot[(Y'[N][0] + I k Y[N][0] == 2 I k) /. sol, {N, 0}];
  Print[Ns]]