I have tried the suggested solution and have not found it possible to disable the headphone jack without also disabling the speakers! My headphone jack is similarly broken, except that the end of the hadphone has snapped off and has jammed inside, so my computed thinks there are headphones connected despite this not being the case. When I go on playback audio devices it lets me test the sound, but doesn't give options on what to do if that sound isn't playing. I need audio to be able to use my laptop for its purpose of helping me make videos so it is very frustrating that it does not seem possible to seperate the speakers from the headphones, considering those are pretty different devices!
Any help would be greatly appreciated, as I am a student and have a deadline looming but can't currently do the work on my computer!