EDIT: After great answer the @vadim-ashikhman. The end of the question.
Resources I'm using:
I managed to add categories and subcategories without any problem.

I have also been able to show the menu with all categories with nestable jQuery library.
So far so good.

Now, when I try to sort through drag and drop. Captured the entire array, here's an example:
[{
  "id": 1,
  "children": [{
    "id": 2,
    "children": [{
      "id": 3
    }, {
      "id": 6
    }, {
      "id": 5
    }, {
      "id": 7
    }, {
      "id": 9
    }]
  }, {
    "id": 8,
    "children": [{
      "id": 10,
      "children": [{
        "id": 11
      }, {
        "id": 12
      }, {
        "id": 13
      }, {
        "id": 14
      }, {
        "id": 15
      }]
    }, {
      "id": 16,
      "children": [{
        "id": 17
      }, {
        "id": 18
      }, {
        "id": 19
      }, {
        "id": 20
      }]
    }]
  }, {
    "id": 22,
    "children": [{
      "id": 23
    }, {
      "id": 24
    }, {
      "id": 25
    }, {
      "id": 26
    }]
  }, {
    "id": 27,
    "children": [{
      "id": 28
    }, {
      "id": 29
    }, {
      "id": 30
    }, {
      "id": 31
    }]
  }, {
    "id": 32
  }, {
    "id": 39
  }, {
    "id": 40
  }]
}]
But not how to update the positions, this has me baffled, I took nearly 16 hours running tests but without any result.
Without any favorable result.
So I come to you to see if you can help me.
EDIT: After great answer the @vadim-ashikhman
$('#nestable').nestable({
          group: 1,
          maxDepth :6
      }).on('dragEnd', function(event, item, source, destination, position) {
	    // Make an ajax request to persist move on database
	    // here you can pass item-id, source-id, destination-id and position index to the server
	    // ....
	    var parent_id = $(item).parent().parent().data('idcata');
	    var actual_id = $(item).data('idcata');
	    var prev_id   = $(item).prev("li").data('idcata');
	    var page_id   = $(item).data('pagina-id');
	    console.log("id "+ actual_id + "\nParent: "+ parent_id +"\nPosition:" + position + "\nPrev : " +  prev_id + "\nPagina_id: "+page_id);
	    $.ajax({
                type: "POST",
                dataType: "json",
                url: '<?=site_url("admin/categories/ordenar")?>',
                data: {
                  id:actual_id,
                  parent_id:parent_id,
                  position:position,
                  prev_id:prev_id,
                  page_id:page_id,
                },
                cache: false,
                success: function(data) {
                    if(data.data==1)
                        alert('Guardado!!');
                    else
                       alert('No se ha podido guardar la posición');
    
                   },
                   error: function() {
                    alert('No se ha podido guardar la posición');
                  }
          });
	});
    $idcata      = $this->input->post('id');
	$newParentId = $this->input->post('parent_id');
	$newPosition = $this->input->post('position');
	$prevId      = $this->input->post('prev_id');
	$page_id     = $this->input->post('page_id');
	$this->nested_set = new Nested_set();
	$this->nested_set->setControlParams('nested_set_tree');
	$categoria = $this->nested_set->getNodeFromId($idcata);
	$categoriaPadre = $this->nested_set->getNodeFromId($newParentId);
	if($newPosition == 0 ){
		$newCategoria = $this->nested_set->setNodeAsFirstChild($categoria,$categoriaPadre);
	}else{
		$prevCategoria = $this->nested_set->getNodeFromId($prevId);
		$newCategoria  = $this->nested_set->setNodeAsNextSibling($categoria,$prevCategoria);
	}
TADA!!! And it works perfectly
When you move a node to new position, get new position previous node and parent node:
setNodeAsNextSibling() method from the library.You dont need to capture and update the entire tree.
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