I see two problems,
first, the alga tabs killed the algae but left the dead algae in the tank to break down and did not solve the problem which is to much nutrient and light.
Second, the slime is either bacteria or possobly mold feeding on the dead algae and producing nitrates.
IMO you need to stop dosing the tank and cure the problem, not treat the symptoms.
Gravel vac the hell out of the tank and do a couple water changes, you might also add 1 tbs of aquarium salt per 5 gallons water while you are at it. Next add some fast growing plants like java moss, java fern, or elodea. And last evaluate how much light the tank is getting, especially direct sun, then sheild it from the sunlight with drapes or tank background. this should keep the slime from coming back, prevent (or at least slow the growth of algae, and if you keep up on water changes/gravel vacs and disrding old growth ont the plants it will also pull down your nitrates.