The following steps revived an iPhone5: a) put your iPhone into DFU mode (Default Firmware Update), then b) restore the iPhone completely. The complete restore process in DFU mode may need to be repeated 5-6 or more time before the iPhone revives. (I'd give up after 10 tries)
NOTE: I would not recommend paying for a battery replacement. Usually it is a software issue. Once an iPhone's battery has been discharged completely, the software date and other parameters seem to get lost, preventing iPhone from properly functioning.