Is there an exact solution for induced velocity during vortex ring state?
By its nature, VRS is not constant over time or location on the rotor disk. Any estimation of induced velocity in VRS is going to be time- and/or location-averaged, this way we can use the same expressions for rotor power and so forth. Well, also in momentum theory, induced velocity is sort of averaged over the rotor disk; there are factors that account for non-uniformities in induced velocity along the rotor radius, but these are relatively simple.
The unsteady nature of VRS changes rotor geometry and this renders any estimation for exact induced velocity invalid. The change/flex in rotor geometry is significant even for one-piece rotors (think multicopters); with hinged rotors, the rotor blade dynamics need to be at the center of any analysis on induced velocity in VRS.
In my experience with a case study with Eurocopter X3, estimations for VRS work OK for high-solidity and rigid rotors; such as fixed-wing propellers in reverse thrust and the X3's propeller generating reverse thrust when the aircraft is in slow forward flight.