One simple way to pull the new image into your cluster is to overwrite the latest tag, specify imagePullPolicy: Always in your deployment and then use kubectl rollout restart deployment my-static-site from within your pipeline. Kubernetes will then terminate all pods and replace them with new ones that pull the latest image.
You can also work with versioned tags and kubectl set image deployment/my-static-site site=my/image:version. This might be a bit nicer and allows imagePullPolicy: IfNotPresent, but you have to pass your version number into your pipeline somehow, e.g. with git tags.