Broken Xbox Wireless headset microphone boom connector.

This is the headset i still have the button and a screw that came off.

Do you have a headset where this worked before? Or did you change headsets and since then the Microphone is muted?

There are two types of 3.5mm headphone jacks called OMTP and CTIA. On both standards, the microphone connector is switched with the ground connector. So if you have an OMTP headset, the controller is listening for a microphone on the ground connector, while
the microphone sends signals to the controller ground.

Did the headset microphone work before? Or is that a new headset, and it has never worked ever?

What kind of headset is your son using? Is it an original Xbox chat headset with a wide connector that goes into the USB-like connector, or are you using a 3.5mm jack headset with three rings on the connector?

You could make it work, but unless your new headset has metal leads coming out of the 2 outer prongs on the connector (the middle is the 1/8th" male jack), you shouldn't do this.

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Just chip off the 2 outer prongs if they are PLASTIC ONLY like on the original boom microphone headsets (the one that had the connector like yours now). It should fit without them. If they have metal on the prongs DO NOT DO IT or you probably lose functionality
all together with the headset.

Be aware that there are two standards for 3.5mm headset connectors. CTIA and OMTP. The Xbox requires the CTIA standard. If your headset is an OMTP headset, the connectors for ground and microphone are reversed and will not work ok the Xbox.

If your headset is OMTP, get an adapter cable for CTIA to OMTP and vice versa.