We present a method for the accurate quantum treatment of the planar three-body Coulomb problem under electromagnetic driving. Our ab initio approach combines Floquet theory, complex dilation, and the representation of the Hamiltonian in suitably chosen coordinates, without adjustable parameters. The resulting complex-symmetric sparse banded generalized eigenvalue problem of rather high dimension is solved using advanced techniques of parallel programming. This theoretical and numerical machinery is employed to provide a complete description of the bound and of the doubly excited spectrum of the field-free two-dimensional (2D) helium atom. For the driven atom, we focus on the near resonantly driven frozen planet configuration, and give evidence for the existence of nondispersive two-electron wave packets.