Depth completion, the task of reconstructing dense depth maps from sparse depth and RGB images, plays a critical role in 3D scene understanding. However, existing methods often struggle to recover high-frequency details, such as regions with fine structures or weak signals, since depth sensors may fail to capture accurate depth maps in those regions, leading to imperfect supervision ground truth. To overcome this limitation, it is essential to introduce an alternative training source for the models. Emerging depth foundation models excel at producing high-frequency details from RGB images, yet their depth maps suffer from inconsistent scaling. Therefore, we propose a novel teacher-student framework that enhances depth completion by distilling high-frequency knowledge from depth foundation models across multiple scales. Our approach introduces two key innovations: Adaptive Local Wavelet Decomposition, which dynamically adjusts wavelet decomposition level based on local complexity for efficient feature extraction, and Topological Constraints, which apply persistent homology to enforce structural coherence and suppress spurious depth edges. Experiment results demonstrate that our method outperforms state-of-the-art methods, preserving high-frequency details and overall depth fidelity.