You can use commercial insecticidal soap (much more effective and safer than dish detergent), Neem oil, vegetable based horticultural oils, Bt-k (Bacillus thuringiensis Kurstaki), and several other commercial products that are deemed appropriate for organic gardeners. I ASSURE you that professional growers who are certified organic growers do not make elixirs out of stuff in their pantries. You can find lots of very good and safe products that are manufactured for the 'organic' market.

I know that there are scads of recipes on line, but I would rather use something that is proven to be affective, that is not likely to harm my plants, and has directions on the label. But that's just me. ;-)

Regarding the Dawn. It is considered one of the most likely of the detergents to cause plant tissue damage due to the additives (fragrance, grease busters, emulsifiers, etc.) It is not 'organic'.

I'd think that an elixir with Jalapeno peppers in it would require fresh vegetables, not canned.

Most all of the dishwashing liquids out there today are detergents, not soaps. Since the active ingrediant in Insecticidal Soap is the fatty acids formed in soap during the chemical reaction that happens during soap manufacture, and there is no such thing in detergents, using detergents will not be effective.
The commercial Insecticidal Soaps have about 1 percent of these fatty acids and if you get real soap and mix some in water (1 teaspoon per quart of water) you will get the same thing.
Mixing together a spray of Garlic and hot peppers, which is an insect deterent with some Insecticidal Soap is not a good thing to do, because they work to opposite ends. One recipe for an insect repellent is to mix one garlic bulb with one medium onion and one, or three, hot peppers (fresh not canned) in a blender with one cup of water. Then allow that mixture to steep for at least 24 hours and mix a teaspoon of that, strained, with 1 quart of water to spray on the plants that need protection. In this case the soap would be a wetting agent and not an insecticide but it is not necessary anyway.