Hi,
If the battery voltage is OK then clean the battery connectors. Check what happens when you connect it to the PC. Connect the device until you hold the Back key (down left key). If the windows see the device in the "Device manager" then may be you have a software problem.
Try a Hard reset. Press and hold left and right buttons under the screen. Press and release the Power button. Follow the instruction for a hard / factory reset.
!!! This will erase all your data !!!
Next, you can try SW update with the PC Companion or directly with the Sony Update Service.
If you haven't reaction from the windows for connected device then yo have a hardware problem.
Regards.