That's very bad news. What did the Apple folks tell you? Was it a faulty battery? I noticed very fast drain - and not because of stale processes consuming 100% CPU time. (I routinely check my JB'n iDevices with top to find stale processes / daemons using the CPU.) Or, was it overheating? I keep my iPhone5 in an Otterbox Defender (with the original screen protector). Given that there was quite warm in the gym (around 28C degrees) and I've kept the iPhone in my hand half of the time further heating the device, this may have contributed to the battery dying on me - dunno.
BTW, I'm still trying to "kick in" the iPhone to save my JB. Yesterday, I got very desperate and tried to restore when the phone was detected by iTunes - was presented the 1600 error. Keeping it on a charger (with different, all-official, cables) didn't help. Now I'm trying to completely drain the battery (with the device in DFU mode with the backlight on) to see whether it helps.
I did the same. Got the 1600 errors on iTunes and couldn't get it back on.
So I took it to the apple store and said I did an update and I can't get the phone back on. They took it in the back and in about 10-15 min brought a replacement. I had the iPhone 5 still under warranty. Sucks that I lost my jailbreak, but couldn't do without a phone for that long.