I have an iPhone 5s, purchased in ~2013. The battery is original and has never been replaced. I have had charging problems with it recently - I needed to wiggle the cable to get it to charge. I thought it was a bad cable.
A couple of days ago, I let the phone discharge and turn off to calibrate the battery. I let it sit until today, when I tried to charge it. When I plugged it in, all it did was flash the low battery symbol with the lightning bolt and lightning cable (not the solid low battery symbol).
I did blow into the lightning port to clean it out. The cable fits snugly, and the pins on the inside of the port are clean. There's no lint or anything in the port. I've tried a couple different cables and chargers (I tried a Linux computer, a MacBook Pro, a wall adapter, and three different cables), and they all do the same thing. iTunes did not detect the phone on the MacBook Pro.
The symbol reappears for a few seconds whenever the phone is disconnected, so I know the phone detects that it's plugged in. I did get the solid low battery symbol to appear once, but unfortunately, I unplugged it because I wanted to use a wall charger.
I will need the phone in less than a week, so any help would be greatly appreciated. Here are some photos: