Correct, Kubernetes is not responsible for the nodes. I would build a health check into your Autoscale Group (I don't know exactly how to do this on AWS, but am happy to show you an example on GCP - aronchick (at) google).

If you can't get to the machine, there are a million reasons why this would be the case - but ssh is a totally separate process, it's way outside of Kubernetes. VERY commonly, you've run out of memory and processes are fighting among themselves (especially since EVERYTHING seems to be failing), but this is total speculation. OS issues are common too - I've spun up clusters switching from one distro to another, same config, and everything worked great.

Disclosure: I work at Google on Kubernetes.