I'm not an apple expert or anything, but after hearing this from a coworker (and doing a quick google search) many of the generic chargers do not work in the 5s (especially after the most recent update.) I'm thinking this may be the problem in your case...the charger is not working so the phone just needs a good charge from either an apple certified charger or a generic that will actually work with the phone. I've read some people are able to pull the cable out, turn it around and then plug it back in facing the opposite direction to get it to work. Many have reported, however, that even when this remedies the cable issue, its still won't charge the phone while it is completely dead (which is likely the case since it won't power on at all.) So if you haven't fixed your problem by now, go buy a new charger and see what happens. You might be able to go into and apple store and borrow a charger there for a few minutes to see if this actually IS the problem. That way you don't waste your money before buying.