Stuck in headphone mode

Try plugging the headphone plug (cable end) into and out of the headphone jack on the iPad multiple times (20 or so at a time, multiple times) rapidly. Also try twisting the headphone plug multiple times while it's plugged into the jack on the iPad and plugging unplugging the headphones. The internal contacts on the headphone jack can get dirty/oxidized and doing the above can help clean off the oxidation. The goal is to try to clean the internal contacts on the iPad's headphone jack. There is an internal switch on the iPad's headphone jack which cuts the sound to the iPad's speakers and tells the unit that headphones are plugged in. This internal switch can get dirty / oxidized causing the symptoms you are seeing. If this doesn't help, contact Apple Support:

Official Apple Support