Locating at-risk workers in hospitals using legacy private 5G networks is a daunting task that involves solving the problem of indoor localization using commercial off-the-shelf proprietary hardware. Currently, no full-stack schemes or realistic indoor positioning experiments have been conducted using 5G. In this study, we present the first comprehensive 5G framework that combines fingerprinting with the 3GPP Enhanced Cell ID (E-CID) approach. Our methodology consists of a machine-learning model to deduce the user's position by comparing the signal strength received from the User Equipment (UE) with a reference radio power map. This challenging method has four main contributions. First, the 3GPP protocols and functions are extended to provide open, secure, and universal core network-based localization functions. Second, to generate a reference map, the first paradigm of Optical Radio Power Estimation using Light Analysis (ORPELA) is introduced. Real-world experiments prove that it is reproducible and more accurate than state-of-the-art radio-planning software. Third, machine-learning models are designed, trained, and optimized for an ultra-challenging radio context. Finally, an extensive experimental campaign is conducted to demonstrate the expected indoor localization performance of realistic 5G private networks.