Battery is totally kaput and won't take a charge possibly? If so, try a new battery. If the phone is completely dead and unresponsive, you won't be able to do anything otherwise. The S4 is quite an old phone now, and if the original battery has been sitting around for six months or more and it was discharged, that could of finished it completely.
Deep discharge over a long term can damage LiPo batteries. If you have a multimeter handy, put it across the battery terminals, and if it reads like 0volts, then that could be the problem, the battery is very likely kaput, and just won't charge.