Which app distribution method I must use for this app?
The iOS Enterprise agreement has the following clause:


  Distribute Provisioning Profiles only to Your Employees and only in conjunction with Your 
  Internal Use Applications for the purpose of developing and testing Your Internal Use 
  Applications; 


In addition it states:


  Allow Your Customers to use Your Internal Use Applications
  on Deployme
  nt Devices
  , but only 
  (i) on Your physical premises and/or on Your Permitted Entity’s physical premises, or (ii) in other 
  locations, provided all such use is under the direct supervision and physical control of Your 
  Employees or Permitted Users (e.g., a sal
  es presentation to a Customer)


Hence for your user base, it would would not be right to distribute apps with enterprise license. Distribute the app through the app store with some kind of authentication mechanism.

You have also host some kind of registration mechanism on your intranet, where the employees would generate user IDs and passwords for themselves and their families.