I had the same problem (although, the circumstances preceding it were not the same as OP's). I was eventually able to get it back working.
While using it, my 5s suddenly shut down without any reason (didn't drop it, overheat it, etc). It was almost on a full charge at the time. I tried everything to get it to power back on, nothing worked. Hard reset, or connecting the wall charger (and leaving it connected for hours) had absolutely no effect. Calling my own number went straight to voicemail. The phone literally looked 100% dead.
Connecting it to iTunes offered a glimpse of hope: iTunes detected an iPhone in recovery mode, offered to Restore it. OK, so it's not "dead"-dead, after all. Halfway through recovery, the screen turns on showing the Apple logo (on black background). Encouraging. "Waiting for iPhone" , and a few minutes later - error 4005. Bummer. Rinse and repeat. Reconnect USB, go through the iTunes restore steps again. Tried this probably 20 times, on 3 PC's with 3 different Lightning cables. Same error 4005. Clearly this is going nowhere. It's either a hardware failure, or comatose software.
Last resort - a "true" power cycle. I pry open the case, disconnect the battery connector, reconnect it a few seconds later, and press the Power button. Lo and behold. The phone powers up, and the data on it is unchanged. It simply works again.
I haven't got a clue as to what caused the problem in the first place. I'm not excluding a hardware failure, but the phone's been running just fine since then (almost a week now), so it could've been a software issue? (granted, software issues may arise as mere symptoms of actual hardware defects)
Anyway, worth a shot...
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