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 as I understand the claim there this series equals $\frac{1}{3}\prod_{k\geq 1} (1-\frac{x^2}{\lambda_k^2})$. I have not verified this claim, but if it's true, then your approach does make sense. **UPD.** Precise argument can be seen in [this answer](https://math.stackexchange.com/q/84326).

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