What is the difference between 嘗試，試圖，and 企圖?
嘗試 means simply "to try" or "have tried". For example, 


  "Try something new" => 嘗試新事物


試圖 is synonymous with 打算, and means "planning to do something". It can be translated as "try" or "attempt", but implies more of a plan. For example, 


  "Trying to write a book." => 試圖寫本書.


企圖 has the same meaning with 試圖, but with a negative connotation. For example, 


  "Planning to rob the bank." => 企圖搶銀行


For your case, both 她 嘗試 / 試圖 修復被撕毀的照片 can be acceptable. If she has not yet begun working on the photo, 試圖 sounds most natural. If she's already in the act, I would suggest  正試圖 or 正在嘗試. If she has already finished, go with 嘗試.