I use 1 to 1.5 tablespoons of Dawn Ultra per gallon of water and it has always worked for me. If you try the Great Value brand or castile soap then it will have different chemicals or more water so you would have to change the dose or that brand may not even work. I'm not sure about a different type of soap but I think Oliver is saying to use a brush and brush a bar of soap and then brush the insect.
I have also used sevin dust liquid and malathion before and it killed them but I do my best to not have to use those things. Sevin dust (the actual dust) may work better because the dust will lay there and continue to kill if you can get it applied to the underside of the leaves.
I'm pretty sure none of the insecticides work on their eggs so if you spray and kill the live insects, the eggs hatch later and it appears that nothing has happened. You should spray, then later spray again to kill the eggs that hatched. Timing is key.
Here is something that says aphid eggs take 7-8 days to hatch and they can be born pregnant, which I didn't know, so again timing of the spray is very important if you have a problem like you have.