The neutron-9 Li interaction and the corresponding low-energy 10 Li spectrum are decisive for the properties of 11 Li described as a three-body system (n + n + 9 Li). We compute structure and breakup properties of 11 Li as function of this interaction. The hyperfine structure due to the spin 3/2 of both 9 Li and 11 Li is needed and treated with special care. We use the hyperspherical adiabatic expansion of the Faddeev equations for the structure and the participant-spectator model for the breakup reactions of 11 Li. We use established experimental constraints of both 11 Li (binding energy, size and differential breakup cross sections) and of 10 Li (a virtual s state below 50 keV and a p resonance around 0.54 MeV). (Bound and virtual states have positive and negative imaginary wave numbers, respectively (zero real parts). Thus the virtual s-state energies should strictly speaking be negative, but we shall here and throughout the paper omit the minus sign and use the absolute value.) Another p resonance must then be present below 0.54 MeV and another s-level must be present between about 0.5 and 1 MeV depending on unknown spin assignments. All established facts are in agreement with our predictions obtained within the same consistent model.