Jailbreaking now is FAR too much of a security risk to recommend to anyone. It was (moderate) fun while it lasted ...
A specific feature of iOS 8.1.1 was to improve performance on older iPhones, so that's the first step I would suggest. The second step would be to check through the system preferences carefully and see if Bluetooth, location, background updating and other battery-draining items in iOS 8 were turned on when it was installed. I now BT, which I previously kept off, turned itself on with iOS 8 (because of the OS X continuity features, which I **LOVE**).
A quick way to test whether its you or the battery dying is to charge it fully, put the iPhone in Airplane mode for a day, see how much has been lost at the end of the day (should be next to nothing, maybe 10 percent if you use the screen/apps a lot).