Send data from textbox with ajax to asp.net mvc 5 controller

I need some help. I write little app using ASP.NET MVC5 with JavaScript, jQuery, Ajax... and I can't send data from javascript to MVC Controller and change the model.

ViewModel

public class MyViewModel
{
//...
    public string SearchString { get; set; }
    public int? FirstInt { get; set; }
    public int? SecondInt { get; set; }
}

Javascript

    function keystroke() {
        var a = 0, b = 0;
        $('#search').keyup(function (event) { a = 1; });

        $('#search').keydown(function (event) { b = 1; });

        $("#search").keypress(function (event) {
            if (e.which === 13) {
                e.preventDefault();
                $('form').click(function () {
                    sendForm(a, b);
                });
            }
        });
    };
    function sendForm(a, b) {
        $.ajax({
            url: @Url.Action("Index", "Home"),
            type: 'POST',
            contentType: 'application/json',
            data: JSON.stringify({
                FirstInt: a,
                SecondInt: b
            }),
            success: function () {
                alert('success');
            }
        });
    };

View

@using (Html.BeginForm("Index", "Home", FormMethod.Post, new { @class = "form-inline", role = "form" }))
            {
                <div class="form-group has-feedback">
                    @Html.TextBox("SearchString", ViewBag.SearchFilter as string, new
            {
                @class = "form-control",
                onclick="keystroke()",
                id = "search"
            })
                </div>
            }

Controller

public async Task<ActionResult> Index(MyViewModel model)
        {
            //...
            if (model.SearchString != null)
            {
                //...
                var a = model.FirstInt;
                var b = model.SecondInt;
            }
            //...
            return View(model);
        }

Help me, please, to send all the values to controller. Those that changed in JavaScript and what I enter in the textbox. Thanks.