That wasn't the question you asked. You asked why Dani would agree to Xi'ans request. As for IIyana, she'ld do it because in part she wouldn't want to have to deal with a possible Tran-controlled Xi'an again.
This pretty much has nothing to do with a ritual about fighting to be reborn,or twisting it to become what basically amounts to be cosmetic surgery. It's about using it to separate two souls/minds from each other. Hence the Dead Souls tie-in.