No, you don't need to repeat it.
The perfective auxiliary throws us into the near past (have/has) or distant past (had), and the rest of the sentence is clearly locked in that time by context.
By repeating the auxiliary it stresses each activity specifically.
So, "I've written the letter and I've sent it." implies you went to all the effort of writing it, and then all the effort of sending it.
Whereas, "I've written the letter and sent it." implies a more fluid action of writing and sending the letter.
In fact, the more compact you make it, the shorter the action is presented.
So, "I've written and sent the letter." sounds complete and brief.
To make it clearer, I'll implement two past participles (that way we know the 'have' is affecting both of them):
"I've come home and done it already."
'Have' applies to both past participles.
Saying, "I've come home and did it already." sounds unnatural and jarring.