How to fix my iPad stuck in headphones mode?

Welcome!

If the only problem with the iPad is the muted issue, it may be the port itself. Some plugs are too long from tip to "hilt" and can jam the mute switch inside the port in the muted position.

In older Macs, you could look into the empty port for a clue. A red glow deep in the port indicated the mute switch was jammed. I have no idea if that applies to iPads.

Apple Genius bars used to give a dire diagnosis of the device needing expensive parts replacement. However end users found you could insert a plug and GENTLY wiggle it back and forth from different compass positions and often free the stuck mute switch. Easy enough to try if you use a light touch.

I've only encountered this on computers but it is easy and non-invasive enough to try on an iPad at home. To illustrate the issues, this image shows the plug from a name-brand set of headphones plugged into one of my older Macbook Pros. Note the gap need to prevent jamming the mute switch:

Irony: the plug on a no-name set of external computer speakers I bought at a yards sale for US$1 would work flush-seated without jamming the port.

Try this.

If your iPad won't turn on or is frozen - Apple Support