Some thoughts about who should see your perfect cover letter (to the law firm):

• Write the obvious person with the title — Hiring or Recruiting Coordinator
• Write the lawyer in the targeted practice area.  Here the idea is to find touch points such as the same undergraduate college or law school, same home town, same interests, etc.
• Write the lawyer’s assistant.  Yes, the assistant.  Write a short letter attempting to establish a rapport with the one person who can arrange an in-person meeting.  This step is important.  It may not work in every case but you have nothing to lose by trying.  Separate yourself from the pack!