It needs to be pointed out that the series $\frac{\sin(x)-x\cos(x)}{x^3}$ comes from the [MSE answer](https://math.stackexchange.com/q/2921768) and it equals $\prod_{k\geq 1} (x^2-\lambda_k^2)$. So, the approach you used does make sense.

I've also verified your answer numerically, and it looks fine as well. Namely, the analytical expression gives $\sum_k a_k^{-2} = 0.0973745978\ldots$, while $\sum_{k=1}^{10^5} \frac {\lambda _k ^2 +1}{\lambda _k ^2 (\lambda _k ^2 +2)} = 0.09737358469\ldots$ with the difference $\approx 10^{-6}$.