MacBook Pro Late 2011 replace headphone 3.5 mm jack due to headphone’s head stuck inside jack

It sounds like a hardware problem in the jack. If you can put pressure on the external part of the plug and change it: cause static or breakup it's probably a hardware problem in the jack.

If you had said you only tried one set of headphones then I'd have said it's probably a problem in the headphone plug but if it's multiple phones and speakers then it's probably the jack.

There's a sensor in that jack that monitors if something's in it or not to switch off the internal speakers. The if there's continuous sound, just switching from internal to external the problem may be that sensor. If the sound cuts out from both internal and external once in a while it's probably the jack.

If the MacBook Pro is still under warranty I'd have Apple look at it, they'll swap motherboard to make it right if it's a hardware problem at no cost to you.

Depending on the breakage, you may not be able to control the software if the "jack detection" sensor is intermittent. It will over rule whatever software setting you have made and send output to the headphone jack each time it senses an insertion.

Often taking this to Apple at a genius bar is no cost if they can dislodge the broken piece with experience (and same for reputable repair shops). Also, you might only pay labor if the jack isn't broken. Usually there is more damage since why would a protected piece break off inside a perfectly functioning receptacle, but it's possible you broke the jack mostly or entirely before inserting it.

Give a shot at setting the sound in the System Preferences, but most of the times this happens, you'll need to get a repair if you can't use software to switch the output once and have it stick.