Writing an eye-catching cover letter is helpful, but persistent follow-up is key.  In my experience (both personally and from mentoring), it is not likely that you will hear something positive back from the perspective top law firm after one round of correspondence.  The standard rejection letter probably sent by a non-lawyer, manager type is the common response.

Do not stop after this first round!  Instead, think of good reasons that will allow you to stay on your target’s radar screen.  He or she should hear from you regularly with substantive news — update on grades, paper you have published, recent success at work, etc.  This step is skipped by many people.  I am not sure why.  Thoughts?  It takes extra effort and time and, of course, is no guarantee but why not separate yourself from the crowd?