The advancement of the Internet of Medical Things (IoMT) has rendered the clustering method pivotal in medical data analysis. Nevertheless, the exposure of patient privacy during this process constitutes a significant security concern. To achieve privacy protection for cluster analysis while ensuring the accuracy of clustering results in IoMT, this paper proposes a Differential Privacy-based availability-enhanced $k$-modes Clustering scheme, named DPKC. First, we introduce partition entropy to quantify attribute weights and calculate the initial center based on density with distance. This can mitigate the impact of random initialization, thus improving the accuracy of the clustering results. Second, we introduce cluster weights to calculate the distance between data points and centers during allocation. This adjustment aims to reduce the difference in compactness between clusters. Third, we employ a geometric mechanism to inject noise into the frequency of attribute values, ensuring data privacy during the iteration process. Theoretical analysis proves that DPKC satisfies differential privacy and prevents information disclosure. Experimental results show that DPKC improves the F-measure metric by 7.36% compared to existing algorithms.