"Fordi" is fine, as long as you remember the adverb and the verb need to switch places afterwards so it would be "Hendes eksistens er nødvendig fordi jeg ikke kan leve uden hende". This inversion doesn't happen after "for".
In English, this use of commas sounds unnatural. I remember that Danish comma use is different than English comma use, though. I'm just saying that the preferred translation should have one less comma than it now does.