If you want to make this scheme more safe you can add haproxy layer between keepalived and kube-apiserver.

Just install haproxy package into your system, and add the next configuration into /etc/haproxy/haproxy.cfg file:

defaults
    maxconn 20000
    mode    tcp
    option  dontlognull
    timeout http-request 10s
    timeout queue        1m
    timeout connect      10s
    timeout client       86400s
    timeout server       86400s
    timeout tunnel       86400s
frontend k8s-api
    bind :::8443 v4v6
    mode tcp
    default_backend k8s-api
backend k8s-api
    option  httpchk GET /readyz HTTP/1.0
    option  log-health-checks
    http-check expect status 200
    mode tcp
    balance roundrobin
    default-server verify none check-ssl inter 10s downinter 5s rise 2 fall 2 slowstart 60s maxconn 5000 maxqueue 5000 weight 100
    server node1 10.9.8.11:6443 check
    server node2 10.9.8.12:6443 check
    server node3 10.9.8.13:6443 check

Now you can access kubernetes api on port :8443. So this way keepalived will monitor the cluster IP, and haproxy will monitor the availability of kubernetes api-servers.

Also you have to configure timeouts for the haproxy, otherwise your kubectl exec commands will be canceled after 50 seconds

timeout client          4h
timeout server          4h

Just configure your kubelets and kubectl clients to connect on 8443 port instead 6443, and final scheme will looks like:

If we will have non working api-server on node1:

And if we will shutdown whole node:

Cheers!