Consider the following integral:
$$ I_1(r)=\int_{-r}^{r} \int_{-\sqrt{r-x^2}}^{\sqrt{r-x^2}} \frac{1} {(x^2+y^2) \log^2\big(\frac{2}{\sqrt{x^2+y^2}}\big)}\,dy\,dx.$$
I am not able to evaluate these integral in terms of $r$ explicitly. However, using Matlab, I managed to evaluate the integral for different values of $r$.
The values are as follows:
$ I_1 (0.036)= 1.4283.$
$ I_1 (0.018)= 1.1937.$
$ I_1 (0.009)= 1.0227.$
$ I_1 (0.0045)=0.8920.$
$\vdots$
$ I_1 (0.0011)=0.7048$
The sequence is decreasing as r decreases. What can we say about $$\lim_{r \to 0} I_1 (r)?$$
By a change of variable $(x,y)=\rho(\cos\theta,\sin\theta)$ you get the integral
$$2\pi\int_0^r\frac{d\rho}{\rho\log^2(2/\rho)},$$ which converges only if $0\leq r <2$, in which case the value is $\frac{2\pi}{\log(2/r)}$, (note that $\frac{d}{d\rho}\left[\frac{1}{\log(2/\rho)}\right]=\frac{1}{\rho\log^2(2/\rho)}$) which tends to zero as $r\to 0$, in agreement with your numerical results.