In this study, we propose an image despeckling method based on low-rank Hankel matrix approach and speckle level estimation. Annihilating filter-based low-rank Hankel matrix, so called ALOHA approach is very useful to various areas, such as image inpainting and impulse noise reduction. The proposed method utilizes this approach because it provides high performance in completing irregularly subsampled images. Speckled image are subsampled using patch-based speckle level estimator which selects pixels with low speckle level and abandon the others. The subsampled image is reconstructed using low-rank structured matrix completion. Our experimental results demonstrate that precisely estimated speckle level improves despeckling performance significantly. The accuracy of the proposed speckle estimator validates better despeckling performance of the proposed method compared with conventional despeckling methods.