My iPad is stuck on headphones mode!!!

Reading the troubleshoting staps you already performed (repeatedly inserting and removing the 3.5mm plug may clear the problem), put your device on charge for at least half an hour, then, keeping it plugged in, try a "forced" restart of your iPad, using these instructions:

• On an iPad with a Home button: Press and hold the power button and the Home button at the same time. When the Apple logo appears, release both buttons.
• On other iPad models: In rapid sequence, press and quickly release the volume up button, press and quickly release the volume down button, then press and hold the power button. Only when the Apple logo appears (it might take 10-20 seconds), release that button (ignore all other screens that might appear, the one with the "slider" to turn the device off included).

If the issue persists, I suggest that you make an appointment (*) and have your device inspected at an Apple Store or AASP (Apple Authorized Service Provider) among this list:--> https://locate.apple.com/choose-country-region . After checking it out, they'll give you all the available options.

Regards

Giulio

(*) You can do it:

• Online:--> Genius Bar Reservation and Apple Support Options - Apple;
• Contacting a local Apple Store or AASP among the list provided above;
• Using the "Apple Store" app, available and downloadable from AppStore.