A quick Google reveals this is a surprisingly common problem. Not just plugs - chopsticks sometimes.
If the socket actually has a switch in it which mutes the speakers you will either have to remove the offending object or change the socket, or work out which wires to disconnect or join together to defeat the switch.
I guess you have tried all the usual methods of removing the broken stub of the jack plug.
Fishing it out with a pin or toothpick
Superglueing the other half of the jack that broke off and tying to pull it out.
Superglueing a toothpick to the lodged in piece and trying to pull it out.
Small pointy tweezers
Bending the tip of a small safety pin and sliding the tip down the side until the shaft of the safety pin is parallel to the wall and the tip is sticking in side of the broken jack and pulling outwards
Inserting a small drill bit with a rotating motion to screw it in the stub and then pulling it carefully out
Likewise with a very small self tapping screw
Inserting a flathead nail, flathead side first with a dab of superglue, into the socket so it contacts the stub and when the glue has dried, pull nail and stub out
Make a paper tube and insert into the socket and then poke in a toothpick with superglue on the tip and stick it to the stub. the paper stops the toothpick sticking to the inside of the socket
Generally superglue glue will dry in 2 to 3 seconds, full strength is achieved in 12 hours, and it continues to dry for 48 hours, so don't be too impatient.
If the headphone detection is done in software (You may be able to tell this if the Windows startup sounds come through the speakers even when the headphone socket has a plug in it)
(This works on Dimension 8400 which also uses Audigy) download Creative Audigy Control Panel, unzip to a folder, run ctpanel.exe and go to "Device settings", and uncheck “Mute speakers upon Headphone Jack Detection.”
http://www.geekymedia.com/media/audigy_ctpanel.zip
