It's not that genitive requires ни, no. It's just when you have consecutive negations those не turn to ни and act as conjunctions, rather than particles.
Я не знаю ни тебя, ни его.
Он не умел ни петь, ни плясать.
As for "Не будят душу без тебя". I think you might be thinking that будят is to be, but it is not. It is a future form of the verb "будить", which means to wake someone up. So the author means that she's the only one who can wake his soul up.