Breaking it down by phrases in an overly literal manner to help maintain Japanese grammar as much as possible:
「世界の全てを」 "the whole of the world"
「手にした」 "to have caused it to be of the hand" = "made it intended for your possession"
「として」 "causing it to be such that" = "having it happen that"
「も」"even/including"
So; "Even making it such that you caused the whole world to be in your possession" = "Even if you had everything"
「それが」"that"
「あなたの」"which is of you
「幸せなの?」"is (it) that which is being happiness?"
So; "That, is it that which is your happiness?" = "Is that what makes you happy?"
Your translation was fine. "Would that make you happy" is an acceptable localization of the phrase "Is that what makes you happy", because they convey the same idea of a person becoming happy given they have everything in the world. The English phrasing just focuses on the condition changing their state of being, while the Japanese phrasing asks if the person considers that condition their definition of happiness.