I am having the same problem, it indeed says "headphone" over my volume bar... Even if its hard to make it go away, I really want it to work. HELP!
It's hard to fix because it's a hardware problem. I googled it a few months back when my iPhone did this, and the general consensus seems to be dust in the headphone jack; especially if it has a chance to get damp and cling. I never did hear of a sure way of dealing with it, and I don't recommend sticking stuff down the headphone jack to clean it.
Some people had luck by just plugging and unplugging their headsets repeatedly. And tried this and air (a dry computer air blower). Nothing worked, and then the next morning it was working again. So I don't have a solution for you, nor do I remember anyone posting a solid solution.
Google it, or contact Apple. They may have a suggestion. If the iPad is less than a year old (or you have Apple Care) you're still in warrantee and they may replace it if it can not be fixed.