Doesn't sound like a damaged battery to me. What you describe sounds like the phone can't boot iOS and thus shuts down or restarts, just like a computer would if it crashes during startup.
Are you sure you are following the correct procedure to get into recovery mode?
1. Connect a the charger cable to your computer and the phone. (USB -> Lightning cable)
2. Restart by holding the home button and sleep/wake button for a few seconds
3. When the screen turns off, release the sleep/wake button but keep holding the home button, the screen will most likely remain turned off. Check iTunes on the computer if it is detecting a phone in recovery mode. 4. Restore software using iTunes.
If this doesn't help or work most likely you have a damaged logic board. This could be due to various reasons, most commonly not using proper [http://www.computerhope.com/esd.htm|ESD] protection during repairs. It says in the guide i linked that you might have felt a shock, but this is not true, ESD damage can be caused by shocks to small you can't feel it.