You're getting very sleeping cause I'm pulling the string very slowly

The average tension of the string (with $m$ as the mass of the object, $L$ as the length of the pendulum, $g$ as the gravitational constant and $\theta_o$ as the angular amplitude of oscillation) is:

$\overline{T}=mg\overline{\cos{\theta}} + mL\overline{\dot{\theta}^2} = mg(1-\overline{\theta^2}) + mL\overline{\dot{\theta}^2} = mg(1+\frac{\theta_o^2}{4})$

Use that $\theta(t) = \theta_o \cos(\sqrt{\frac{g}{l}} t+\phi)$ . The oscillation energy and potential energy is $E= -mgL \cos{\theta_o} \approx -mgL(1-\frac{\theta_o^2}{2})$ , and we balance the work done by shorten the pendulum length with the change in the oscillation energy to get:

$dE=-mgdL+\frac12 mg\theta_o^2 dL + mgL \theta_o d\theta_o =dA = -TdL = -mg(1+\frac{\theta_o^2}{4})dL \Rightarrow L d\theta_o = -\frac34 \theta_o dL$

$\Rightarrow \theta_o \sim L^{-3/4}$

So the change in the amplitude must be $k^{3/4} = 1.68$