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We present Hubble Space Telescope optical images, Keck-OSIRIS near-infrared (NIR) integral field spectroscopy data cubes and Keck-Near InfraRed Camera-2 (NIRC2) NIR images of nova V5668 Sgr from 2016 to 2019. The observations indicate enhanced emission at the polar caps and equatorial torus for low-ionization lines, and enhanced high-ionization emission lines only at the polar caps. The radial velocities are compatible with a homogeneous expansion velocity of v = 590 km s−1 and a system inclination angle of 24°. These values were used to estimate an expansion parallax distance of 1200 ± 400 pc. The NIRC2 data indicate the presence of dust in 2016 and 2017, but no dust emission could be detected in 2019. The observational data were used for assembling 3D photoionization models of the ejecta. The model results indicate that the central source has a temperature of 1.88 × 105 K and a luminosity of 1.6 × 1035 erg s−1 in August of 2017 (2.4 yr post eruption), and that the shell has a mass of 6.3 × 10−5 M⊙. The models also suggest anisotropy of the ionizing flux, possibly by the contribution from a luminous accretion disc.