How to insert a span inside a search form?

If we look at the source code of get_search_form(), notice that before the form gets rendered, the search_form_format filter hook gets fired. We can use that to add another filter attached to get_search_form where the callback is dependent upon the format.

add_filter( 'search_form_format', 'wpse_259716_search_form_format', 99, 1 );
function wpse_259716_search_form_format( $format ) {
  if( in_array( $format, array( 'xhtml', 'html5' ) ) ) {
    add_filter( 'get_search_form', "wpse_259716_get_search_form_$format", 99, 1 );
  }
  return $format;
}

function wpse_259716_get_search_form_xhtml( $form ) {
  $search="<input type="submit"";
  $xhtml="some xhtml";
  $replace = $xhtml . $search;
  return str_replace( $search, $replace, $form );
}

function wpse_259716_get_search_form_html5( $form ) {
  $search="<input type="submit"";
  $html5 = 'some html5';
  $replace = $html5 . $search;
  return str_replace( $search, $replace, $form );
}

Alternatively, you could use a class-based approach.

$wpse_259716 = new wpse_259716();
add_filter( 'search_form_format', array( $wpse_259716, 'search_form_format' ), 99, 1 );
add_filter( 'get_search_form', array( $wpse_259716, 'get_search_form' ), 99, 1 );

class wpse_259716 {
  protected $format;
  public function search_form_format( $format ) {
    return $this->format = $format;
  }
  public function get_search_form( $form ) {
    $search = $replace="<input type="submit"";
    if( 'xhtml' === $this->format ) {
      $xhtml="some xhtml";
      $replace = $xhmtl . $search;
    }
    elseif( 'html5' === $this->format ) {
      $html5 = 'some html5';
      $replace = $html5 . $search;
    }
    return str_replace( $search, $replace, $form );
  }
}