A couple of questions and points: if your well is a "deep" well (i.e. similar to what Kenny Mapes described), then sanitizing the well is a non starter. The hydrogen sulfide is in the aquifer. The katalox system is probably designed for that application. It's possible the katalox needs an oxidizer.
Chlorine can be used to "regenerate" and is bactericidal, however some folks may be concerned about disinfection by-products (you would have to make a determination about organic material in water that would react with the chlorine). Other folks use a H2O2 - type oxidizer that is also bactericidal.
Importantly, if you have a conventional water softener (cation exchange resin), then it's important NOT to use bleach to "sterilize" the resin bed because the resin will significantly degrade. I use peracetic acid (you can purchase it at some farm stores that service commercial farmers - it's used as a bactericide/bacteristatic on fruit and produce and to clean packing equipment.
If you do not have a water softener, then the chlorine bleach works well - and is cheap. BTW, you don't need much bleach (do a Google search). Some variables to consider are: volume of water (vessels, piping), presence of organic material (e.g. slime), temperature (Florida - nominal 72 F), dwell time. It's possible that you will need to add a proportional mixer of an oxidizer (chlorine based - NaClO, hydrogen peroxide based) if the problem reoccurs.
Regarding slime in the system, pull the anode in the hot water heater..... Florida water in general has iron and sulfur metabolizing bacteria that feast on the sulfur and iron, and they enjoy bathing in 120 F warm water. There are solutions, but it takes a little thought and coordination.
Do a Google search on: disinfection byproducts in drinking water ( haloacetic acids (HAA) and trihalomethanes (THM)) good luck.