It needs to be pointed out that the series $\frac{\sin(x)-x\cos(x)}{x^3}$ comes from the MSE answer, and as I understand the claim there this series equals $\prod_{k\geq 1} (x^2-\lambda_k^2)$. I have not verified this claim, but if it's true, then your approach does make sense.
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 $\approx 10^{-6}$.