Starting from stationary bifurcations in Couette-Dean flow, we compute stationary nontrivial solutions in the circular Couette geometry for an inertialess Finitely Extensible Nonlinear Elastic (FENE-P) dumbbell fluid. These solutions are isolated from the Couette flow branch, arising at finite amplitude in saddle-node bifurcations as the Weissenberg number increases. Spatially, they are strongly localized axisymmetric vortex pairs embedded in an arbitrarily long ``far field'' of pure Couette flow, similar to experimentally observed ``diwhirl'' patterns (Groisman and Steinberg, 1997, Phys. Rev. Lett., vol. 78, 1460) and ``flame patterns'' (Baumert and Muller, 1999, J. Non-Newtonian Fluid Mech., vol. 83(1--2), 33). In addition, the computed velocity fields show qualitative and quantitative similarities to the experimentally observed diwhirls. For computationally accessible parameter values, these solutions appear only above the linear instability limit, in contrast to the experimental observations. Correspondingly, they are themselves linearly unstable. Extrapolation of the trend in the bifurcation points with increasing polymer extensibility, however, suggests that for sufficiently high extensibility the diwhirls will come into existence before the linear instability, as seen experimentally. Based on the computed stress and velocity fields, we propose a fully nonlinear self-sustaining mechanism for these flows. The mechanism is related to that for viscoelastic Dean flow vortices and arises from a finite amplitude perturbation giving rise to a locally unstable profile of the azimuthal normal stress near the outer cylinder at the symmetry plane of the vortex pair. The unstable profile in combination with a ``tubeless siphon'' effect nonlinearly sustains the patterns. We propose that these solitary, strongly nonlinear structures comprise fundamental building blocks for complex spatiotemporal dynamics in the flow of elastic liquids.