I am interested in the following general double sums, for integers $a\geq 1$ and $b\geq 2$,

$$Z(a,b) = \sum_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^a} \frac{2\ell+3}{(\binom{k+2}{2}+\binom{\ell+2}{2})^b},$$

which are converging very slowly. For these sums, there is also an alternative expression as an iterated integral in dimension $a+b$, similar to multiple zeta values.

I would like more particularly to find $Z(2,2)$ and $Z(1,3)$ with precision as large as possible. Using the naive summation, I could only obtain 4 decimal digits, namely $Z(2,2) \simeq 4.7058$ and $Z(1,3) \simeq 1.6470$.

It is known that $2 Z(2,2) + 4 Z(1,3) = 16$.

What would be a smart way to accelerate the convergence, in general and in the special case using maybe the previous formula ?