If the system indicates a set of headphones is connected, then that would cause the regular speakers not to work as it should be sending the audio out of the headphone jack. I suspect the headphone pin detector is either broken or stuck in the "on" position. You can try examining the headphone port to see if the pin is visible. If it is in a disengaged position, you should be able to see the pin just barely visible in the jack. If it's in an engaged position, it may be difficult to see.
Some have reported registry edits to bypass jack detection and others have stated inserting and removing headphones multiple times eventually got the pin to disengage. However, these are not true fixes and would typically require a motherboard replacement to properly correct the issue. With that being said, you could try the steps listed here (untested to verify if it works) if you have access to regedit or know someone in IT that does. Most businesses typically disable regedit for various reasons. I would also highly recommend creating a restore point or a backup of the registry saved to an external storage device in the event something goes in a different direction.