How to add a conditional billing field based in Country in "Commerce Kickstart"? I added a new billing mandatory field called "NIN Number" at admin/commerce/customer-profiles/types/billing/fields

But I only need to show this new field only if the selected country is United Kingdom

How can I do this? 

Will the https://drupal.org/project/conditional_fields module be usefull to get it?
I don't think you can use conditional fields because of the internal structure of the addressfield module, you need to use some custom code here, using the #states property and checking against the value of the country. 
Using hook_form_alter, you can edit that form, here's an example of #states coming from the examples module:

  $form['high_school'] = array(
    '#type' =&gt; 'fieldset',
    '#title' =&gt; t('High School Information'),
    // This #states rule says that the "high school" fieldset should only
    // be shown if the "student_type" form element is set to "High School".
    '#states' =&gt; array(
      'visible' =&gt; array(
        ':input[name="student_type"]' =&gt; array('value' =&gt; 'high_school'),
      ),
    ),
  );