If it has vertical bars running through the picture, your LCD is damaged and the only thing that will fix it is replacing the front assembly.
Iphone 5 front assembly costs $30 with all tools included, and takes 10-15 minutes.... maybe an hour max if its you're first time.
If everything else on the phone is working properly, replacing the front assembly should solve your issues.
Water damage is a tricky thing, results and symptoms differ widely
If replacing the front assembly doesn't help, I'd disassemble the phone completely and thoroughly scrub all the corrosion from the components with 90%+ isopropyl alcohol. Wouldn't hurt to submerge the logic board in the alcohol and shake it up a bit either.
If the alcohol bath doesn't work, I wouldn't dump anymore time or money into it.