Iphone 5s stuck in headphone mode
If there is nothing physically jammed into the headphone jack then it should be easily fixable. 

This happens to me occasionally when sound refuses to play out the external speakers and the simplest solution is to restart your phone by holding down the sleep button until it says slide to power off. Then, restart it by either plugging it in to a power source or holding the sleep button for an extended few seconds.

On the other hand if this doesn't work and the piece logged in there is the problem, try cleaning out the jack using things such as tweezers or compressed air. If all else fails, turn it in the the nearest Apple Store ( via in person or by mail ) and have them take a look at the device in person.

Hope this helped!