I had the same problem and finally got it fixed last night. The spring goes in with the slightly longer end with one bend first. This goes in a slot toward the bottom and has to hook around something inside. I put it in with the flash closed but kept checking until I had that end right. To do this I opened the flash while holding the spring until I finally had it so that end of the spring was staying in the slot and not visible. This took several tries. Then I carefully closed the flash while holding the spring and re-attached the top end. For a LONG time I had been trying to just attach this end by turning it counterclockwise and of course it wouldn't stay in the hole. It has to be turned clockwise (going over the top of the spring) and then into it's hole. I hope my answer is helpful to others with this problem.
All this time I was trying to put the bracket right next to the battery but that's not right. There's no way the bumps on the metal connector can fit in the slots on the bracket. It's battery, metal connector and then the bracket. So easy. Next time I dismantle an iPhone 5S I won't be dreading this step.
I had to replace the power flex cable, which means taking almost everything out. Scary! When I restarted, my power button and flash were fixed but my volume was the same as yours and my home button didn't work. :( That Touch ID cable and cover are from h#$l! Anyway, I went back in and everything is working now. Instead of doing them like in the photos and videos, I attached the silence button first and then the volume buttons. So when you put in the middle screw that holds them both in, the screw goes through volume first and then silence.
All this time I was trying to put the bracket right next to the battery but that's not right. There's no way the bumps on the metal connector can fit in the slots on the bracket. It's battery, metal connector and then the bracket. So easy. Next time I dismantle an iPhone 5S I won't be dreading this step.
I had to replace the power flex cable, which means taking almost everything out. Scary! When I restarted, my power button and flash were fixed but my volume was the same as yours and my home button didn't work. :( That Touch ID cable and cover are from h#$l! Anyway, I went back in and everything is working now. Instead of doing them like in the photos and videos, I attached the silence button first and then the volume buttons. So when you put in the middle screw that holds them both in, the screw goes through volume first and then silence.
Oh ya, ifixit is a great resource. Thanks!