Is my translation right?
As @S.Rhee has pointed out, and IMHO, below are some issues with your translation:


影响 translates to "affect/influence".
安全 in this context translates to "security".
职责 translates better to "duties/obligations".
The second "that" modifying clause should modify 职责, not 国家安全情报.


A more accurate translation might be,


  "To carry out the other duties, as indicated by the President or the Director of National Intelligence, that are related to influencing the National Security Intelligence."