I messed you up more then I helped, sorry man. I need my notes/book/calculator and they are all in my car.. sorry to take you down the wrong track like that
565 is the force upward! it is the Normal force (Fn). the sumation of all forces on her is zero (she is not falling ie. accelerating down, or up) so Fn=Fg and as you said Fg=565N=Fn
does the problem look like this?
A --------- c ----------------- B
---------- D
AB would be the rope, D would be the climber, 80* would be the angle between Dc and DB, and 65* would be the angle between Dc and DA?