General steps: update Kafka's advertised.listeners and make sure Kafka is accessible (e.g. allow inputs on Security Groups).
You will probably face the following problems.
If you want to connect to Kafka outside of EKS, you will need to change advertised.listeners. When a client connects to a single Kafka bootstrap server (like other brokers), a bootstrap server sends a list of addresses for all brokers to the client. If you want to connect to a EKS Kafka, these default values will not be correct. This post provides an excellent explanation of the field.
If you use Helm to deploy Kafka to AWS EKS, please review the chart's README. It describes multiple setups for communicating into EKS.
Running helm upgrade on the Kafka chart does not always update the pods. The exact reason is unknown. It's probably an issue with the chart's implementation. You should run kubectl describe pod and other commands to see the current status of the pods. During initial development, you can run helm uninstall and then helm installto force the values to update.