Don unfortunately it sounds like to me you are going to need to open it up and check the battery itself to ensure it is securely connected as well as the charging port connector. There is a metal cover that holds both down but they have been known to pop loose. If you are a do it yourself type it is not real difficult you could try it yourself but you will need some specialty tools and there are plenty of videos on youtube that take you step by step through the process. If you are not comfortable I would take it into a local repair shop and let them take a look. Good luck!
Don even though it may seem aggravating this is what the website is about, sharing knowledge and troubleshooting things down until you find the problem. The 100% issue is a software glitch that has affected the iPhone 6 the most but it has affected the 5 series as well. Lets cover the battery first so you know the battery is dead because you read it out with a meter so I would agree you need to order a new battery and I would order from this website because the knock off cheap batteries have known to cause issues and not have the battery life. Its my opinion that the battery is acting as a ground allowing the phone to boot up via the wall plug so even though it is dead it is allowing the phone to obtain ground therefore booting the phone. If you install a new battery and see the false 100% there are two ways to try and reset the software. You can do a forced reset by holding the home and power buttons at the same time until the phone turns off and then the apple logo is shown, at this point let go of both buttons. The other is more in depth and requires you to back up the phone, do a DFU reset which will completely erase the phone. The reason you do this is to reinstall the software and firmware with the most up to date through iTunes. The DFU reset is the last resort typically the forced restart will correct the issue. If you get to that point let me know and I will walk you through it but for now try the battery replacement.