I'm no expert on water softeners but from what I understand the water does not pass through the salt tank but through a mineral tank filled with small polystyrene beads, also known as zeolites. At timed intervals a liquid brine solution is pumped through the mineral tank.The sodium ions in the brine replace the magnesium and calcium that attached them selves to the zeolites.
It's most likely the membrane in the mineral tank failed and it's the zeolites that's in your system.
I've had to back flush a system. I disconnected water heater supplies and connected a hose to the hot supply ( not the heater) then cap off the cold supply to heater or turn off supply valve to heater. Cap off a tub or shower head and turn on that valve to back feed cold water into the hot water system and out the hose on the wtr htr supply. Do that to all the fixtures if you can. Sometime you have to actually take off the angle stop at the fixture and back flush it with a hose from a clean source.
Here'a link I found on softeners How a water softener works