A detailed analysis of a coronal loop oscillation event is presented, using data from the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory (SDO) for the first time. The loop oscillation event occurred on 2010 October 16, 19:05–19:35 UT and was triggered by an M2.9 GOES-class flare, located inside a highly inclined cone of a narrow-angle coronal mass ejection. This oscillation event had a number of unusual features: (1) excitation of kink-mode oscillations in vertical polarization (in the loop plane), (2) coupled cross-sectional and density oscillations with identical periods, (3) no detectable kink amplitude damping over the observed duration of four kink-mode periods (P=6.3 minutes), (4) multi-loop oscillations with slightly (≈10%) different periods, and (5) a relatively cool loop temperature of T ≈ 0.5 MK. We employ a novel method of deriving the electron density ratio external and internal to the oscillating loop from the ratio of Alfvénic speeds deduced from the flare trigger delay and the kink-mode period, i.e., ne/ni = (vA/vAe)2 = 0.08 ± 0.01. The coupling of the kink mode and cross-sectional oscillations can be explained as a consequence of the loop length variation in the vertical polarization mode. We determine the exact footpoint locations and loop length with stereoscopic triangulation using STEREO/EUVI/A data. We model the magnetic field in the oscillating loop using Helioseismic and Magnetic Imager/SDO magnetogram data and a potential-field model and find agreement with the seismological value of the magnetic field, Bkink = 4.0 ± 0.7 G, within a factor of two.