NOTE: I don't quite get the same answer, but it's out by just a small fraction. I can't find any mistakes here, but that certainly doesn't rule it out... (Headbang) Nonetheless, I hope the methodology will at least be useful! lol
Split the series into two:
$$\sum_{k=0}^{\infty}\frac{(-1)^k}{(3k+2)^2} - \sum_{k=0}^{\infty}\frac{(-1)^k}{(3k+1)^2}$$
And then split each of those into two, by splitting the index k into odd and even terms 2k and 2k+1, thus eliminating the alternating signs:$$\sum_{k=0}^{\infty}\frac{1}{(6k+2)^2} - \sum_{k=0}^{\infty}\frac{1}{(6k+5)^2} -
\sum_{k=0}^{\infty}\frac{1}{(6k+1)^2} + \sum_{k=0}^{\infty}\frac{1}{(6k+4)^2}=$$
$$-\sum_{k=0}^{\infty} \left[ \frac{1}{(6k+1)^2} + \frac{1}{(6k+2)^2} + \frac{1}{(6k+3)^2} + \frac{1}{(6k+4)^2} + \frac{1}{(6k+5)^2 } + \frac{1}{(6k+6)^2}
\right] +$$
$$2\, \sum_{k=0}^{\infty}\frac{1}{(6k+2)^2} +
\sum_{k=0}^{\infty}\frac{1}{(6k+3)^2} +
2\, \sum_{k=0}^{\infty}\frac{1}{(6k+4)^2} +
\sum_{k=0}^{\infty}\frac{1}{(6k+6)^2}=$$
$$-\sum_{k=1}^{\infty}\frac{1}{k^2} + 2\, \sum_{k=0}^{\infty}\frac{1}{(6k+2)^2} +
\sum_{k=0}^{\infty}\frac{1}{(6k+3)^2} +
2\, \sum_{k=0}^{\infty}\frac{1}{(6k+4)^2} +
\sum_{k=0}^{\infty}\frac{1}{(6k+6)^2}=$$
$$-\zeta(2) + 2\, \sum_{k=0}^{\infty}\frac{1}{(6k+2)^2} +
\sum_{k=0}^{\infty}\frac{1}{(6k+3)^2} +
2\, \sum_{k=0}^{\infty}\frac{1}{(6k+4)^2} +
\sum_{k=0}^{\infty}\frac{1}{(6k+6)^2}=$$
$$-\zeta(2) +2\, \sum_{k=0}^{\infty} \left[ \frac{1}{(6k+2)^2} + \frac{1}{(6k+4)^2}\right] +
\sum_{k=0}^{\infty} \left[ \frac{1}{(6k+3)^2} + \frac{1}{(6k+6)^2}\right] = $$
$$-\zeta(2) +\frac{2}{2^2}\, \sum_{k=0}^{\infty} \left[ \frac{1}{(3k+1)^2} + \frac{1}{(3k+2)^2}\right] +
\frac{1}{3^2}\, \sum_{k=0}^{\infty} \left[ \frac{1}{(3k+1)^2} + \frac{1}{(3k+2)^2}\right] =$$
$$-\zeta(2) +\frac{11}{18}\, \sum_{k=0}^{\infty} \left[ \frac{1}{(3k+1)^2} + \frac{1}{(3k+2)^2}\right] = $$
$$-\zeta(2) +\frac{11}{18}\, \sum_{k=0}^{\infty} \left[ \frac{1}{(3k+1)^2} + \frac{1}{(3k+2)^2} + \frac{1}{(3k+3)^2} \right] - \frac{11}{18}\, \sum_{k=0}^{\infty} \frac{1}{(3k+3)^2} = $$
$$-\zeta(2) +\frac{11}{18}\, \sum_{k=1}^{\infty}\frac{1}{k^2} - \frac{11}{18}\, \frac{1}{3^2}\, \sum_{k=0}^{\infty}\frac{1}{(k+1)^2} =$$$$-\zeta(2) + \frac{11}{18}\, \zeta(2) -\frac{11}{162}\, \zeta(2) = -\frac{74}{162}\, \zeta(2) = $$$$-\frac{74}{162}\, \frac{\pi^2}{6} = -\frac{37\pi^2}{486}
$$
NOTE:
$$\frac{2}{27} = 0.0740740740...$$
$$\frac{37}{486} = 0.076131687...$$Looks like I need help on this one too! (Sun)