You could clean the corrosion off the motherboard by using a tooth brush and isopropyl alcohol but by the look of the damage it's probably best just to replace the charging block as there isn't much damage on the actual motherboard only the part.
NOTE: You may have to wait 30 seconds-1 minute when performing each step. If you don't want to lose DATA there may be a "Forgot password?" option in which case google will reset the password via your email. If this option is not there then follow these steps. To perform factory/hard reset please follow those steps (you will lose everything): 1. Turn the power off. 2. Press and Hold the Volume Up and Power button at the same time. 3. When the start up screen show (Samsung logo displays on the screen), release all the buttons. 4. Use the Volume Up/Down button to navigate to "wipe data/factory reset", select with Power button 5. Select "wipe data/factory reset" 6. Select "Yes -- delete all user data" (1) Allow several minutes for the factory data reset process to complete. 7. Select "reboot system now" (1) Allow several minutes for the reboot process to complete.
When a phone comes in contact with water you cannot get rid of the damage unless you have taken the phone apart. When the phone is in use the water travels faster (it's kinda like Osmosis) so this will take the water through the motherboard. Then the heat and working of the electronics/components in the phone causes it to corrode the motherboard so if you were to open up the phone it would look kind of like green moss around the connections and it can also cause the phone to lose the use of components and lose solder which obviously is not what you want. The rice and hair dryer tricks are always used but usually don't work. If you want to restore the life of the battery you'll need the following (along with the necessary tools): 1.New battery. 2.Toothbrush. 3.Isopropyl Alcohol. 4.Goggles. 5.Nose and mouth mask. 6.Latex/Plastic Gloves. 7.Eraser. Strip the phone and take out the motherboard. Using gloves and goggles and possibly a mask (Isopropyl is very bad when breathed in especially for your lungs). Dip...
When iPhone's are plugged into the mains they do not need battery life for them to show the Apple logo, so you should be seeing the Apple logo when it is plugged in. This means that the phone is not recognizing the lightning cable, try a new cable. It is a common fault on all of the iPhone 5 series that the charging ports go I replace too many a day/week so I would suggest just replacing the charging port as it's most likely to be that plus they don't cost too much.
If you want to restore the life of the battery you'll need the following (along with the necessary tools): 1.New battery. 2.Toothbrush. 3.Isopropyl Alcohol. 4.Goggles. 5.Nose and mouth mask. 6.Latex/Plastic Gloves. 7.Eraser. Strip the phone and take out the motherboard. Using gloves and goggles and possibly a mask (Isopropyl is very bad when breathed in especially for your lungs). Dip the toothbrush in Isopropyl Alcohol and clean the board with it taking away all of the corrosion. On the gold contact plates on the board use the eraser to get the dirt/corrosion off them do not use the toothbrush as it could scratch and cause damage. Hopefully if all goes well this should work you may also need to reflow the motherboard to set everything again. There will most probably be parts that need changing but you'll know that when the device is on. After you've cleaned it wait 20-30 minutes for the board to dry (it needs to be completely dry otherwise this could cause damage). Check the board for any black bits that...
You may just need to restore the iPhone using iTunes if this doesn't work it's most probably your digitizer (the thing that controls the touch). The flex may be disconnected or you may just need a new screen. Use this guide for help.
As Kyle said with the water damage use Isopropyl Alcohol to clean the corrosion. You may then want to replace the battery of the phone. With it vibrating when turning on it's most likely to be your LCD and digitizer that have gone try replacing your screen.