A 3.5mm jack broke into my mac's jack input and I can't take it out (we tried for a day, it's absolutely impossible) and because of that my MacBook detects a plugged in audio output. I can't listen music or audio output on my Mac's speakers.
Is there a way to force the sound to come out from speakers, even while the system detects a plugged in jack?
This is a hardware problem, unfortunately. The problem here is that inside the port there is a "switch" that detects whether a jack is plugged in or not and depending if one is, plays audio through the jack or the internal speakers.
The linked article talks about a malfunctining switch/port, but this isn't your case as you described; you have a stuck 3.5mm jack that won't come out so the switch is set to play audio through the port and unfortunately, there's no overriding it.
The only other options are to:
get a USB DAC (Digital Audio Converter) which is basically a USB soundcard to hook your audio/speakers up to
take the Mac in for service to have a new audio port soldered on.
Someone has had a very similar case Headphone jack broke and got stuck in headphone socket MacBook Pro
It was also solved through hardware though may be possible through software if someone knows how.
Allan's answer is correct - this is a hardware switch, you can't get the built-in speakers back while that bit is stuck in there.
However, I would like to add a simpler suggestion, since I had this exact same thing happen to me.
Buy an external USB speaker - these can be gotten very cheaply (though perhaps not for the greatest quality).
Just plug in the USB speaker, then go to System Preferences > Sound > Output and select the speaker there. You may have to redo this every time you restart the computer, but this worked fine for me with no need for additional ports or soundcards.
This also worked for me with a USB headphone jack as well - I used my computer for several months like this with the bit of headphone still stuck in the audio jack.
