Not sure if this will be of help for anyone. But I had a similar thing happen to my wifes iphone 4. I replaced the old cracked screen with a new one and carefully re assembled the phone back together. Once the battery was connected, I turned the phone back on to see if the screen worked. At this point everything looked fine. Battery was at 16% as well. So i put the phone on the charger to charge over night and it showing showing that it was 100% charged. I didn't think much of this. So basically in the morning once i took it off the charger it died straight away.
Spent all day trying to find forums to help me pin point the issue. Tried charging the phone again and all it would do was turn on with the apple icon and stay on that screen for 3mins before it made it to the main home screen. then after 2mins it would turn off and restart itself. So i had a feeling that it could've been a battery issue. The battery itself i'm pretty sure was long due to die.
So i managed to figure out what the problem was. I took my iphone and I decided to try my battery in my wifes iphone and took her battery from her phone and tried it on mine. Turns out that my battery worked fine on my wifes iphone while her battery didn't turn my phone on. So the issue was a dead battery.
So with the battery that wouldn't work...i took a closer look at the ribbon connector and it look like it was bent to the point of breaking off. So if you think, you put everything back together carefully, then I would check the battery to make sure that the ribbon connector isnt damaged etc.