Dualshock 4 won't turn on outside of steam anymore

I suspect you've lost the pairing with your PS4 system. According to the Sony instructions, you need to pair it by plugging it directly in to the PS4 with a USB cable. Have you tried this, you don't specifically state that in your question?

http://manuals.playstation.net/document/gb/ps4/basic/usecontroller.html