Many problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary (IB) method. In the IB approach to such problems, the elastic forces generated by the immersed structure are applied to the surrounding fluid, and the motion of the immersed structure is determined by the local motion of the fluid. Recently, the IB method has been extended to treat more general elasticity models that include both positional and rotational degrees of freedom. For such models, force and torque must both be applied to the fluid. The positional degrees of freedom of the immersed structure move according to the local linear velocity of the fluid, whereas the rotational degrees of freedom move according to the local angular velocity. This paper introduces a spatially adaptive, formally second-order accurate version of this generalized immersed boundary method. We use this adaptive scheme to simulate the dynamics of an elastic ring immersed in fluid. To describe the elasticity of the ring, we use an unconstrained version of Kirchhoff rod theory. We demonstrate empirically that our numerical scheme yields essentially second-order convergence rates when applied to such problems. We also study dynamical instabilities of such fluid-structure systems, and we compare numerical results produced by our method to classical analytic results from elastic rod theory.