If you haven't, you'd better define a lenght() function so given one linked list you can know how many nodes does it have.
Thanks to the response of Cereal_Killer to the previous version of this answer, I noticed that the list is firstly a singly linked list, and you just have to make the middle node be linked both to the next node and to some previous node.
Now I guess that you have defined two structures (Struct, Class or whatever depending on the language you're using). So lets say you have Node_s defined as a node with only a next pointer, and Node_d with both a next and a prev pointer. (Node_d may inherite from Node_s so you just have to add the prev attribute in the child class). Knowing this, the code above should be doing what you need:
function do_it(my_LinkedList linkedList){
int i_middle;
int length = linkedList.length();
if ( (length ÷ 2 ) != 0 ) i_middle = length / 2;
else return -1;
Node_s aux = linkedList.first();
int index = 0;
Node_d middle= null;
while (aux != null) {
if (index == i_middle - 1){ //now aux is the middle's previous node
middle.data = aux.next.data; //aux.next is the middle singly node, we assignate the data to the new double linked node
middle.prev = aux; //as we said, aux is the prev to the middle
midle.next = aux.next.next; //so aux.next.next is the next to the middle
print(what you need to print);
}else {
print("Node " + index " next: "+ aux.next);
}//end if
index++;
aux = aux.next;
} //end while
}//end function
This previous code should be doing what you need. I wrote the answer in some kind of pseudo-java code so if you're not familiar with Java or don't understand what my pseudo-code does, please let me know. Anyway, the idea of my code may present some troubles depending on the language you're working with, so you'll have to adapt it.
Note that at the end of the execution of this program, your data structure won't be a singly linked list, and neither a double one, since you'll have linkedList.length() - 1 nodes linked in a signly way but the middle one will have two links.
Hope this helps.
prev(3)= 2?interfacetype, so a previous or next node could point to either a singly or doubly linked list node. I would run from this company asking such a silly question... run very fast.