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The ubiquity, large bandwidth, and spatial diversity of the fifth generation (5G) cellular signal render it a promising candidate for accurate positioning in indoor environments where the global navigation satellite system (GNSS) signal is absent. In this paper, a joint angle and delay estimation (JADE) scheme is designed for 5G picocell base stations (gNBs) which addresses two crucial issues to make it both effective and efficient in realistic indoor environments. Firstly, the direction-dependence of the array modeling error for picocell gNB as well as its impact on JADE is revealed. This error is mitigated by fitting the array response measurements to a vector-valued function and pre-calibrating the ideal steering-vector with the fitted function. Secondly, based on the deployment reality that 5G picocell gNBs only have a small-scale antenna array but have a large signal bandwidth, the proposed scheme decouples the estimation of time-of-arrival (TOA) and direction-of-arrival (DOA) to reduce the huge complexity induced by two-dimensional joint processing. It employs the iterative-adaptive-approach (IAA) to resolve multipath signals in the TOA domain, followed by a conventional beamformer (CBF) to retrieve the desired line-of-sight DOA. By further exploiting a dimension-reducing pre-processing module and accelerating spectrum computing by fast Fourier transforms, an efficient implementation is achieved for real-time JADE. Numerical simulations demonstrate the superiority of the proposed method in terms of DOA estimation accuracy. Field tests show that a triangulation positioning error of 0.44 m is achieved for 90% cases using only DOAs estimated at two separated receiving points.