Unfortunately, you might not have a case.
Per IATA guidelines, interline partners have upto 24 hours to confirm/deny interline inventory requests. When being endorsed to another carrier during IRROPS, standard practice is for the endorsing carrier to confirm inventory with the operating carrier via EDIFACT message or via telephone.
If the DL agent that endorsed your ticket did not follow this procedure, KL may have denied DL the inventory, which would have put you on standby status. Unfortunately, no recourse in these situations, although you may be able to request a customer service gesture from DL for improperly rebooking you.
That said, if the agent that rebooked you at JAX did get approval from KL, then you have a pretty good case for IDB, in my opinion.