We propose a lattice-based scheme for secret key generation from Gaussian sources in the presence of an eavesdropper, and show that it achieves the strong secret key capacity in the case of degraded source models, as well as the optimal secret key / public communication rate trade-off. The key ingredients of our scheme are the use of the modulo lattice operation to extract the channel intrinsic randomness, based on the notion of flatness factor, together with a randomized lattice quantization technique to quantize the continuous source. Compared to previous works, we introduce two new notions of flatness factor based on $L^1$ distance and KL divergence, respectively, which might be of independent interest. We prove the existence of secrecy-good lattices under $L^1$ distance and KL divergence, whose $L^1$ and KL flatness factors vanish for volume-to-noise ratios up to $2πe$. This improves upon the volume-to-noise ratio threshold $2π$ of the $L^{\infty}$ flatness factor.