According to Borwein, page 356 Prop. 2,
$\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$
holds for $k\in\left(0,1\right]$.
$I(1,k)=K(k')$ is the Complete Elliptic Integral of the first kind (Wikipedia).
I need to find a similar estimate for complex numbers but Borwein uses Mean Value Theorem, which is not applicable for complex numbers.
To find something similar I think one has to restrict oneself to complex numbers with positiv real part and absolute value smaller than 1.
Would be great, if somebody provided a different approach or idea!