I just bought an adapter for the Microsoft controller that allows for chat.
Is this what you have?

Your mixamp should include a diagram of how to connect it. There are a few different ways to do it and I don't know what cables you have. For example here are two ways (there are more)
You can use the Y cable that was included with your stereo adapter and connect the headset side to the mixamp and then connect your mic to the mic side of the Y adapter. Connect your headset to the mixamp Y cable.
or
You can connect a TRRS 3.5mm M to 2.5mm M cable from the stereo adapter to the mixamp and then plug your mic in to the Y cable that is attached to your mixamp input ports.
You might have better luck asking someone who has a mixamp how to set it up if your mixamp didn't include a diagram.