Is there a Kibana health API for load balancer?

Elastic Stack Kibana
2

Depends on what you want to achieve. If you just want to let the LB know if the node is OK, check the status returned by /status:

kibana-b :: ~ »  curl -I  http://127.0.0.1:5601/status
HTTP/1.1 200 OK
kbn-name: kibana
kbn-version: 6.2.3
...

If you want detailed status, just omit the token:

kibana-b :: ~ » curl -s http://127.0.0.1:5601/api/status | json_pp
{
   "version" : {
      "number" : "6.2.3",
      "build_snapshot" : false,
      "build_number" : 16602,
      "build_hash" : "2582ab6060209a9fc6a67275fc52ce08ae74c957"
   },
   "status" : {
      "overall" : {
         "since" : "2018-03-21T09:52:41.957Z",
         "nickname" : "Looking good",
         "state" : "green",
         "title" : "Green",
         "icon" : "success"
      },
      "statuses" : [
         {
            "message" : "Ready",
            "since" : "2018-03-21T09:52:41.957Z",
            "id" : "plugin:kibana@6.2.3",
            "state" : "green",
            "icon" : "success"
         },
...