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We present a reanalysis of reverberation-mapping data from 2005 for the Seyfert galaxy NGC 4151, supplemented with additional data from the literature to constrain the continuum variations over a significantly longer baseline than the original monitoring program. Modeling of the continuum light curve and the velocity-resolved variations across the H$β$ emission line constrains the geometry and kinematics of the broad line region (BLR). The BLR is well described by a very thick disk with similar opening angle ($θ_o \approx 57^{\circ}$) and inclination angle ($θ_i \approx 58^{\circ}$), suggesting that our sight line towards the innermost central engine skims just above the surface of the BLR. The inclination is consistent with constraints from geometric modeling of the narrow line region, and the similarity between the inclination and opening angles is intriguing given previous studies of NGC 4151 that suggest BLR gas has been observed temporarily eclipsing the X-ray source. The BLR kinematics are dominated by eccentric bound orbits, with $\sim10$% of the orbits preferring near-circular motions. With the BLR geometry and kinematics constrained, the models provide an independent and direct black hole mass measurement of $\log M_{\rm BH}/M_{\odot} = 7.22^{+0.11}_{-0.10}$ or $M_{\rm BH}=1.66^{+0.48}_{-0.34}\times10^7 M_{\odot}$, which is in good agreement with mass measurements from stellar dynamical modeling and gas dynamical modeling. NGC 4151 is one of the few nearby broad-lined Seyferts where the black hole mass may be measured via multiple independent techniques, and it provides an important test case for investigating potential systematics that could affect the black hole mass scales used in the local Universe and for high-redshift quasars.