Hey,
1st
Maybe try downloading f0recast-v1.3 it displays all the current info about your iPhone but it waits for your phone to connect. if it does not then it may be a dead phone.
2nd option
Change the battery. but purchase a legit OEM battery
they are more expensive but well worth it.
3rd option
Try looking at the bottom connector if any dirt is trapped then clean. or replace the dock connector , not expensive.
4th option
Try the reset 10 seconds power down with holding the home button. keep doing it till something happens might not work though.. but may be that your battery just gave out. so needs replacing