Counts of $k$-cliques are commonly used metrics in subgraph mining. Since graphs often have sensitive data, there also has been a lot of work on $k$-clique counts with differential privacy. However, these metrics have very high global sensitivity, and so more sophisticated techniques have been developed for counting $k$-cliques with privacy. Smooth sensitivity and ladder functions were developed for reducing the noise magnitude for private estimates of these metrics. However, these are computationally very inefficient to estimate. No polynomial time algorithms are known for smooth sensitivity of $k$-cliques for $k>3$, while the time complexity of ladder functions is lower bounded by the time for exact counts, which does not scale very well. In this paper, we develop a new highly scalable algorithm for estimating $k$-clique counts with differential privacy. Our algorithm adapts the ladder function to serve as a smooth upper bound on its local sensitivity, and utilizes the approximation sensitivity framework to calibrate noise with magnitude proportional to an approximation of the bound. This gives us a significant improvement in the running time. Experiments show that our method is several orders of magnitude faster than the ladder function based estimates of $k$-clique counts, while the accuracy is similar. Our algorithm is the first to scale to graphs with millions of edges, and for larger $k$, for which the ladder function algorithm doesn't complete.

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