Unobtrusive check for Inline Datepicker does not work, works fine for non-inline Datepicker

I did not use jquery.ui datepicker because I needed to select a whole week and several dates under different conditions, so I use Keith Wood's datepicker in the EditorTemplate field for Date fields:

    @model Nullable<DateTime>
    @{
        string name = ViewData.TemplateInfo.HtmlFieldPrefix;
        string id = name.Replace(".", "_");
        string dt = Model.HasValue ? String.Format("{0:d}",(string)Model.Value.ToShortDateString()) : string.Empty;
    }        
    @Html.TextBox("", dt, new { @class = "datefield", @type = "date", @id = id })
    <script type="text/javascript">
        $(document).ready(function () {
            $('#@id').datepick({
                renderer: $.datepick.themeRollerRenderer,
                multiSelect: 999
            });
        });    
    </script>

The check works correctly when using a text field and has a datepicker popup; but I really want to use the built-in datepicker, but I can’t get confirmation to work in these conditions:

    @model Nullable<DateTime>
    @{
        string name = ViewData.TemplateInfo.HtmlFieldPrefix;
        string id = name.Replace(".", "_");
        string dt = Model.HasValue ? String.Format("{0:d}",(string)Model.Value.ToShortDateString()) : string.Empty;
    }

    <div class="kwdp">
    </div>
    @Html.Hidden("", dt, new { @class = "datefield", @id = id })
    <script type="text/javascript">
        $(document).ready(function () {
            $('.kwdp').datepick({
                renderer: $.datepick.themeRollerRenderer,
                multiSelect: 999
            });
        });    
    </script>

Here is an example of my client side custom validation:

    form.validate({ 
        rules:{
            StartDate: { 
                required: true, 
                dpDate: true 
            }
        },
        messages: {
            StartDate: 'Please enter a valid date (dd-mm-yyyy)'
        }
    });

And here is the definition of the DateTime field in my data class:

    [Required]
    [DataType(DataType.Date)]
    [DisplayFormat(ApplyFormatInEditMode = true, DataFormatString = "{0:dd-MM-yyyy}")]
    public Nullable<DateTime> StartDate { get; set; }

, onSelect datepicker, (), , , , . html :

    <input name="StartDate" class="datefield" id="StartDate" type="hidden" data-val-required="The StartDate field is required." data-val="true" data-val-date="The field StartDate must be a date." value=""/>

:

    <script src="@Url.Content("~/Scripts/jquery.validate.min.js")" type="text/javascript"></script>
    <script src="@Url.Content("~/Scripts/jquery.validate.unobtrusive.min.js")" type="text/javascript"></script>
    <script src="@Url.Content("~/Scripts/jquery.unobtrusive-ajax.min.js")" type="text/javascript"></script>    
    <script src="@Url.Content("~/Scripts/datepick/jquery.datepick.min.js")" type="text/javascript"></script>
    <script src="@Url.Content("~/Scripts/datepick/jquery.datepick.ext.min.js")" type="text/javascript"></script>
    <script src="@Url.Content("~/Scripts/datepick/jquery.datepick.validation.min.js")" type="text/javascript"></script>
    <script src="@Url.Content("~/Scripts/jquery.ui.datepicker.validation.min.js")" type="text/javascript"></script>

, , , . , ?