An attack like this doesn't really need the exploited computers to run inference, do they? If I was an LLM bent on destruction of the internet, I'd be writing programs to run on each computer, not turning each computer into an LLM itself.
A few programs to break in, install themselves and remain asleep until they are needed, another few to spread through grabbing every OpenAI, GLM, whatever key, another one to remain asleep on computers (whether hosted or desktops) that have adequate GPU, etc.