I'm not sure if someone have done Datetimepicker in ST2, but have seen couple of those in ST1.
You can have a look into https://github.com/tomalex0/SenchaTo...Datetimepicker.
Screenshots are available at https://github.com/tomalex0/SenchaTo...ree/master/img

I have also setup few examples to get a better idea of implementation of DateTimePicker and DateTimePickerField.

Have a look and let me know any issues.