Hi, welcome to iPF!
Try cleaning the connector. Maybe there's some dust that makes your iPad think the headphones are still plugged in. Be very cautious when doing so. Use a soft brush or do it with a vacuum cleaner.
You could also try to plug in the headphones again, play some music, stop it or pause it, then remove the headphones and force close the app you used to play the music: double tap the Home button to reveal the multitask bar, then (iOS 6: ) tap and hold an app icon until they start to wiggle and delete them; (iOS 7: ) also tap and hold to see the bar, slide pictures above the app icons towards the top of the screen.
Look at your settings: what did you choose in Settings - General - Use Side Switch to. If it's "Mute", change to Lock Rotation, then back again.
Hope that helps.