I used to work at an Apple Store. I was never a tech, so take this with a grain of salt. It's PROBABLY not related to the system update but more than likely something bad on the phone itself. If you see the usage and standby meters the same like you do, you're right, something is hung up in the background. The best thing to do is to make sure you have a backup and restore the phone. If there's something hanging in the background for a while, you'll start to notice your phone acting funny. Like the battery life problem. Left alone for long enough and you'll start getting corrupted data into the phone because of it which will make your phone act really strange. Apps constantly crashing to the point of unusability, it can become unresponsive on certain parts of the screen, etc. And that corrupted data can eventually get copied into the backup file - meaning that any restore from the backup will bring that problem right back.
To Back Up - First copy all photos and videos from the phone to the computer. They will fill up your iCloud space really fast. Then look for Apps that hold documents. Most apps are really just portals to websites so they contain little to no data. However, for apps that contain files, make sure you have those files backed up somewhere. Then backup the phone to either iCloud or by plugging it into iTunes. Plug it into iTunes in either case and also right-click on the phone and hit Transfer Purchases. That will transfer any apps, music, movies, etc that are not currently on the computer from the phone.
So easiest to hardest steps after backing up all data:
1: Reset All Settings: Go to Settings>General>Reset All Content and Settings. Then restore from backup. This can kill a lot of problems and takes maybe 15 minutes. However, if the problem is deep in the system...
2: Restore from Backup: Plug the phone into iTunes and do a System Restore. This will do a clean install and will clear a lot more problems. Then restore from backup. Now, if the problem persists, your backup is more than likely corrupted so...
3: Restore the Phone: Do the same process as above but do not restore from the backup. You can re-sync to iCloud for contacts and re-sync apps from the computer (or re-download them). Essentially you're starting with a fresh system and just copying the apps back over. You'll lose your app data, so maybe some high scores in some games, nothing too bad. However, if the problem persists even after doing this...
4: DFU Restore: Look up how to do this online. If none of the above have fixed it, there's some bad juju on your phone. A DFU will reinstall the system AND the firmware. Just to be on the safe side, I would advise not restoring from backup. If your phone was borked enough that it needed a DFU restore, don't take the chance of putting stuff back on there. Just copy the apps back.
All in all, they're very easy steps to do and when all is said and done, you can have your phone back up and running in about 30 min with any of the above steps. Hope it helps, Cheers!