Carefully remove the joystick from its housing.

There is a thin black gasket around the hole where the joystick pokes through the Joy-Con. Do your best not to disturb this gasket as you remove the joystick!

Once the Joy-Con is fully reassembled, connect it to your Nintendo Switch and calibrate the new joystick. Additionally, you may need to power cycle your Switch by holding down the power button for 10-15 seconds until it turns off, then turning it back on.