At this point I think you'll need to take a good look at the main logic board for corrosion damage.
Disconnect the battery and if needed remove the logic board fully as there is more circuitry on the other side. Don't forget to practice good ESD protection as static can kill your exposed board.
Use a solution of distilled water and baking soda (10 to 1). Let the solution settle or filter it though a coffee filter cone as you don't want the solids. Using cotton swabs (Q-Tips) wipe the areas down and if its stubborn use a toothbrush to scrub lightly (you don't want to pop off any small components brushing to hard). Once you clean off the areas go over the spots again this time with straight distilled water. Then again with good quality of Isopropyl alcohol (85% or better). The distilled water don't have anything in it so it will dry cleanly Vs tap or spring water. The Isopropyl alcohol is to act as a dryer pulling out any remaining water thats under the components. In any case you still want to let the circuit board dry fully for a good day. I like putting in on a sunny window sill so the sun warms it slightly.
Focus on the area nearest the jack connector.
If this doesn't work then I think you'll need to replace the logic board, or get a USB DAC to power a set of external speakers. - Sorry