Searching for the sources of high-energy cosmic particles requires sophisticated analysis techniques, frequently involving hypothesis tests with unbinned log-likelihood (LLH) functions. SkyLLH is an open-source, Python-based software tool to build these LLH functions and perform likelihood-ratio tests. We present a new easy-to-use and modular extension of SkyLLH that allows the user to perform neutrino point source searches in the entire sky using ten years of IceCube public data. To guide the user, SkyLLH provides tutorials showing how to analyze the experimental data and calculate useful statistical quantities. Here we describe the details of the analysis workflow and illustrate some of the possible methods to work with the IceCube public dataset. Additionally, we show that SkyLLH can reproduce the results from a previous IceCube publication that used the public data release. We obtain a similar local significance for the neutrino emission from a list of candidate sources within a maximum shift of 0.5$\sigma$. Finally, the measured neutrino flux from the most significant source candidate, NGC 1068, shows substantial agreement with the previously published result.