Only when lui is used as an indirect object pronoun can it mean him or her. e.g. "Je lui donne le livre." -> I give him/her the book.
Otherwise, lui always means him, and elle always means her. e.g. "Je marche jusqu'à lui." -> I walk to him. "Je marche jusqu'à elle." -> I walk to her.
"Jusque" means "until", "to", but usually, you don´t use that word alone but with another preposition: à, that´s why you have "jusqu'à" (a contraction of jusque + à).
"Jusque" can be used before "assez", "aussi", "fort", "bien", "si", "très" but it's mostly used in literature. In ordinary French "jusqu'à" is commonly used.
I'm going to suggest that it is because jusqu'a doesn't mean "to", it means "up to" or "until" and not in the sense that "I walk up to you and slap you in the face" but rather "I walk up to the end of the road and turn around." If you were to say "I walk to her" I believe you'd say "Je marche à elle"
But I could be wrong