Jetpack already includes a “date” field, but it is not part of the form builder. You’ll need to add the field into the shortcode directly, like so:
[contact-form][contact-field label='Name' type='name' required='1'/][contact-field label='Email' type='email' required='1'/][contact-field label='Website' type='url'/][contact-field label='Comment' type='textarea' required='1'/][contact-field label='Pick a date' type='date'/][/contact-form]
The Time and File input options do not exist yet, however. You can follow our progress on adding more options to the Contact Form builder here:
https://github.com/Automattic/jetpack/issues/269