The limit in question is equal to
$$\def\tfrac#1#2{{\textstyle\frac{#1}{#2}}}
\tfrac14\pi\log\left(\frac{16\pi^3e^{2\gamma}}{\Gamma(\frac14)^4}\right)
-G-\zeta(2)
\\ =
-0.82586\ 11759\ 78831\ 08201\ 02008\ 35613\ 80953\ 63017\ 94512\
34066\ 96955\ 08772 \ldots $$
in terms of Euler's $\gamma$ and Catalan $G$.
Demonstration.
The sum
$$ S(N) = \sum_{1\leq m,n\leq N} \frac1{m^2+n^2} \sim \tfrac12\pi\log
N $$
is in an inconvenient form, because it is taken over the square
$[1,N]^2$ instead of the circle of radius $N$:
$$ S_2(N) = \sum_{m,n\geq 1}\frac{[m^2+n^2\leq N^2]}{m^2+n^2} \sim
\tfrac12\pi\log N. $$
This is so, because the circle-sum can be rewritten as a
one-dimensional sum involving the function $r_2(k)$ that counts the
number of ways to write $k$ as a sum of two squares of (signed)
integers. So we can write the sum $S_2$ as a quarter of the sum over a
circle in $\mathbb{Z}^2$ excluding the axes, like so:
$$\begin{eqnarray}
S_2(N) &=& \frac14 \sum_{m,n\in\mathbb{Z}}' \frac{[m^2+n^2\leq
N^2]}{m^2+n^2} - H_N^{(2)}
\\&=& \frac14\sum_{1\leq n\leq N^2} \frac{r_2(n)}{n} - H_N^{(2)}.
\end{eqnarray} $$
Here $H_N^{(2)}$ has the limit $\zeta(2)=\frac{\pi^2}{6}$, and the
prime above the sum means the sum omits $(m,n)=(0,0)$.
The sum
$$ g(n) = \sum_{1\leq k\leq n} \frac{r_2(k)}{k} = \pi\log N + \pi S +
O(N^{-1/2}) $$
has a standard asymptotic form in terms of the
Sierpinski constant
$$ S = \log
\left(\frac{4\pi^3e^{2\gamma}}{\Gamma(\frac14)^4}\right). $$
The difference $S(N)-S_2(N)$ between the circle-sum and the square-sum
does not diverge as $N\to\infty$, and we may sandwich it above and
below with two integrals that converge to the same value of
$$ \int_1^N dx\int_1^N dy \frac{[x^2+y^2>N^2]}{x^2+y^2}. $$
This integral can be computed in polar coordinates, by writing
$$ \begin{eqnarray}
2\int_{N}^{N\sqrt{2}}\frac{dr}{r} \int_0^{\pi/4}d\phi\,[r\cos\phi <
N]
&=&
2\int_1^{\sqrt{2}}\frac{dr}{r}\left(\frac\pi4 - \arccos\frac1r\right)
\\ &=& \tfrac12\pi\log2 - G.
\end{eqnarray}$$
Putting everything together leads to the expression given above.