This problem is usually due to an attempt to aggressively unscrew the dome from the stick. The dome is held onto the stick using an adhesive (something akin to Locktite) that is supposed to keep the dome from unscrewing during normal use. Users can attempt to unscrew the dome using the longer thumbstick option available in the accessory pack. When attempting to unscrew the dome using this method, do not twist with a lot of force. If too much force is applied the stick itself may break inside the stick housing, leading to the the issue the OP is having. If the stick does not twist off fairly easily, then use a hair drier or a hot air station set to low heat (120C or lower) to warm the dome, and also loosen the adhesive. Then twist and it should release and twist off.
If the stick is damaged due to over-torquing the dome, it will need to be replaced. In that case, follow the guide available on ifixit. The part can also be acquired form ifixit or Microsoft directly.
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