My built-in audio output doesn’t work

It's entirely likely you have a situation where the drivers on Windows are not detecting a faulty hardware condition and thus continue to send signal to the copper ports. Your Mac may be sending the audio to the TOSlink (optical) portion of the jack.

See this other question for details and a similar answer by @chico on his experience getting around this hardware failure by using different OS / drivers.

Your speakers not working after you unplug your headphone jack is a dead giveaway that your headphone jack is full of dust or some other kind of foreign substance. Get a can of compressed air (you must be 18 to purchase compressed air, also it costs about $5 per can) from Wal-Mart or Best Buy or another retailer and use it to blow the headphone jack clean and you should find that it works again.