Chosen Solution
Usually, when the headphone is removed, the device stops playing music. However, lately, the device keeps playing music even after the headphone is removed -and the battery get drained.
First, try unplugging the headphone jack from the logic board and plugging it back in. If that doesn’t work, replace the headphone jack. iPod 5th Generation (Video) Headphone Jack & Hold Switch Replacement Of course, you could damage the iPod more while attempting a repair, so if you’re not 100% confident, get used to pausing your music after you unplug your headphones, which is probably what I would do, to be on the safe side.