Hi Lauren, sorry to hear about the phone. The best thing to do when a phone comes in contact with water is to power off the phone, disassemble it as soon as possible and clean its logic board and any parts that are involved with its circuitry. Use 90-99% isopropyl alcohol to clean the logic board and other parts. I use a tooth brush or similar softer bristle brush to scrub all the parts down. The reason water is bad for the circuitry is because when it dries it leaves mineral deposits behind that corrode and eat away at the metal components causing poor electrical connections and more problems. If left uncleaned it will keep corroding and other parts of the phone may start to fail too. Isopropyl alcohol(90%+) helps break up any corrosion and it evaporates very quickly. You could probably find a video tutorial on the internet to show how this is done.
The thing that scares me about this situation is that he left the phone on, and then plug it into a charger. Hopefully it was not still wet inside the phone when it was charging (which I am guessing it was). The sooner the battery is turned off and removed from the phone the less likely it will short out. Look over the logic board for any obvious burn marks or damage that may signal a short.
I think getting a new battery was a good idea because lots of times that is the first thing to fail. It can be truly difficult to pinpoint the true problem when it comes to water damage. That's why its best to clean it all the best you can then reassemble it. If none of this process helps and it's still not working correctly with the new parts you have purchased I would sadly guess that you mother board has been damaged by an electrical short and may need replaced or repaired. Hopefully not though.
Best of luck, and hopefully the isopropyl alcohol bath will help.