I only know a little on this. As I understand it, the controller should suck out an amount during each regeneration. The solution sucked out goes out of the drain line of the controller -- the same line that carries the backwash water.
If your solution is not being sucked out, you need to give your controller some attention. Find the service manual. Clean the injector screen, the injector (which actually is a little venturi pump that does the sucking).
I see water on the floor. Is that from the solution tank being full and the solution coming out from where the tubing comes in?
Somebody with more knowledge will have more help. I am suggesting that you get the service manual for your 5600 (
not sxt, and
not economizer) controller. Start reading.
http://www.softenerparts.com/Fleck_5600_Valve_and_Parts_s/30.htm has a link to the 56oo manual. I think that is what you have, although maybe yours is a variation.
Also, measure your media tank (what the controller sits on). You want diameter and height. 3.5 GPM for the backwash rate suggests that it might be 8" x 44" or less.