jQuery : Split the DOM Node using jQuery

One way to do it would be:

last_text=$('.split')[0].nextSibling; //get textNode after split span
prev_text=$('.split')[0].previousSibling; //get text Node before split span
current=$('.split').text(); //get split span text

$('.second').html(' '); //clear current html


cloned1=$('.start').clone(); // clone el
cloned1.insertAfter($('.start'));
cloned2=$('.start').first().clone(); //clone one more
cloned2.insertAfter($('.start').last());
//set text for elements
$('.start .second').eq(0).html(prev_text);
$('.start .second').eq(1).html(current);
$('.start .second').eq(2).html(last_text);

Demo:

last_text=$('.split')[0].nextSibling; //get textNode after split span
prev_text=$('.split')[0].previousSibling; //get text Node before split span
current=$('.split').text(); //get split span text

$('.second').html(' '); //clear current html


cloned1=$('.start').clone(); // clone el
cloned1.insertAfter($('.start'));
cloned2=$('.start').first().clone(); //clone one more
cloned2.insertAfter($('.start').last());
//set text for elements
$('.start .second').eq(0).html(prev_text);
$('.start .second').eq(1).html(current);
$('.start .second').eq(2).html(last_text);

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<span class="start">
  <span class="second">
   new text <span class='split'>is written</span> in this place.
 </span>
</span>

P.S. If there are no classes (i guess that could be the case), you will have to use different selectors, to target spans (inside container?), with eq(), maybe, but, basically, this simple logic should work...