Laravel 5.2 Dropdown Search Filter

Question

I have a selectbox/dropdown in my sidebar.blade.php View:

    {!! Form::open(array('method'=>'patch','route'=>'search')) !!}
    {!! Form::select('search',$categories,null,array_merge(['class'=>'postform ts-select'],['placeholder'=>'Select Category'],['id'=>'search'],['name'=>'search'])) !!}
    <div class="row">
        <div class="col-xs-12 col-sm-4 col-md-3 col-centered" style="margin-top: 15px">
            {!! Form::submit('Search', array('class'=>'blog-read ts-button')) !!}
        </div>
    </div>
    {!! Form::close() !!}

The select box receives its values from model -> Blog.php :

public static function getCategories()
{
    return self::groupBy('category')->lists('category', 'category');
}

On BlogController.php:

public function getIndex()
{

    $mostRecommended = \App\Blog::mostRecommended();
    $last = \App\Blog::lastPosts();
    $categories = \App\Blog::getCategories();
    //echo'<pre>';
    //dd($mostRecommended);
    return View('blog::index', array('title' => "Welcome ", 'last' => $last, 'mostRecommended' => $mostRecommended, 'categories' => $categories));

}

public function search($category)
{
    $query = Request::get('search');
    $articles = DB::table('blog')->where('category', '=', $query);
    $mostRecommended = \App\Blog::mostRecommended();
    $last = \App\Blog::lastPosts();
    $categories = \App\Blog::getCategories();

    return View('blog::index', array('title' => $query, 'articles' => $articles, 'last' => $last, 'mostRecommended' => $mostRecommended, 'categories' => $categories));
}

Routes:
Route::controller('/blog', '\Serverfireteam\blog\BlogController');

//Below line doesn't work (obviously)
Route::get('/blog/search',['as'=>'search','uses'=>'Serverfireteam\blog\BlogController@search']);

My main content (list of displayed posts) is iterated through the index.blade.php View as:
@foreach($last as $post)
This iterates appropriately for ALL blogs. I need to make this dynamic with the dropdown / select box. I am sure that I need AJAX (but this is my first project and I'm clueless)

Here's a small sample of what I have tried:

Filter with dropdown Laravel

Laravel Ajax dropdown filter

Laravel 5.2 filter with dropdownlist

laravel 5.2 - search function

Get selected text from a drop-down list (select box) using jQuery

I know that I am close and it is my inexperience that's destroying me here.

My question: Please, where am I screwing up? I have spent days on this one issue.

** UPDATE Here, I updated my code to try something else; please take a look below

<div id="categories-6" class="widget widget_categories">
    <div class="title-widget"><h3>Categories</h3></div>
    <label class="screen-reader-text" for="cat">Categories</label>

    {!! Form::open(array('method'=>'patch','route'=>'search')) !!}
    {!! Form::select('search',$categories,null,array_merge(['class'=>'postform ts-select'],['placeholder'=>'Select Category'],['id'=>'search'],['name'=>'search'])) !!}
    <div class="row">
        <div class="col-xs-12 col-sm-4 col-md-3 col-centered" style="margin-top: 15px">
            {!! Form::submit('Search', array('class'=>'blog-read ts-button')) !!}
        </div>
    </div>
    {!! Form::close() !!}

</div>

Here is a script that I currently have on my index.blade.php

<script>

    $('#search').on('change', function () {
        var category = $(this).val();
        var base_url = $({!! json_encode(url('/')) !!}).val;
        $.ajax({
            url: base_url + "/blog/search/" + category,
            dataType : "json",
            success: function (data) {
                $('#inner-container').html(data.html); //here container is the wrapper of index view
            }
        });
    });
</script>

Here is an updated look at my BlogController.php

public function getIndex()
{

    $mostRecommended = \App\Blog::mostRecommended();
    $last = \App\Blog::lastPosts();
    $categories = \App\Blog::getCategories();
    //echo'<pre>';
    //dd($mostRecommended);
    return View('blog::index', array('title' => "Welcome ", 'last' => $last, 'mostRecommended' => $mostRecommended, 'categories' => $categories));

}

Lastly, the routes.php

Route::get('/blog/search/{category}', <br>
['a s'=>'search','uses'=>'Serverfireteam\blog\BlogController@search']);

Here is the error I'm receiving

ErrorException in UrlGenerationException.php line 17:

Missing required parameters for [Route: search] [URI: blog/search/{category}].
(View:web/hearten/vendor/serverfireteam/blog/src/views/sidebar.blade.php)
(View: web/hearten/vendor/serverfireteam/blog/src/views/sidebar.blade.php)
(View: web/hearten/vendor/serverfireteam/blog/src/views/sidebar.blade.php)