Out of the box Kubernetes has no support for assigning an EIP to an AWS Network Load Balancer. There is currently a pull request for this feature listed here: https://github.com/kubernetes/kubernetes/issues/63959#issuecomment-484203661
This feature request has unfortunately not been approved yet.
Not to worry though I have put together a simple application you can run as a CronJob on your current Kubernetes cluster that will clone the targets from your Kubernetes Service LoadBalancer to an AWS Network Load Balancer that you create and have an EIP assigned to.
Docker hub repository: https://hub.docker.com/r/newtondev/k8s-aws-nlb-target-cloner
Source code repository: https://github.com/newtondev/k8s-aws-nlb-target-cloner
Want to learn more about Kubernetes. Please check out this great book that I helped review: