You don’t always respond to emails right away. In fact, sometimes you put them off until the next day, the next week, or—downcast gaze—the next month. At some point the calculus shifts from “Can I somehow compose an email that justifies my glacially slow response-time?” to “Would it be easier to just fake my death instead?”
While it doesn’t look or feel great, sometimes you have to own up to sleeping on someone’s message.