Try putting your input inside of the actual form:
<script type="text/x-kendo-template" id="template">
<div id="details-container">
<h5 data-idtest="1">#= Id #</h5>
<h2>#= TitleDisplay # - #= ArtistDisplay #</h2>
@using (Ajax.BeginForm("TheAction", "Search", new AjaxOptions { UpdateTargetId = "#=Id" }))
{
Input The Day:<input id="TheDayTextBox" name="TheDay" type="text" />
<button class="btn btn-inversea" title="Log outa" type="submit">Log Offsdfsaf</button>
}