Ultrasound imaging provides real-time views of internal organs, which are essential for accurate diagnosis and treatment. However, speckle noise, caused by wave interactions with tissues, creates a grainy texture that hides crucial details. This noise varies with image intensity, which limits the effectiveness of traditional denoising methods. We introduce the Metric-Optimized Knowledge Distillation (MK) model, a deep-learning approach that utilizes Knowledge Distillation (KD) for denoising ultrasound images. Our method transfers knowledge from a high-performing teacher network to a smaller student network designed for this task. By leveraging KD, the model removes speckle noise while preserving key anatomical details needed for accurate diagnosis. A key innovation of our paper is the metric-guided training strategy. We achieve this by repeatedly computing evaluation metrics used to assess our model. Incorporating them into the loss function enables the model to reduce noise and enhance image quality optimally. We evaluate our proposed method against state-of-the-art despeckling techniques, including DNCNN and other recent models. The results demonstrate that our approach performs superior noise reduction and image quality preservation, making it a valuable tool for enhancing the diagnostic utility of ultrasound images.