How to paginate the archives links

Hi!
I am not a guru coder, and I’ve been trying to figure this out all day. How to paginate my archives.php page, so that I only see 10 links of my previous posts per page. I found the paginate_links( $args ) function but nothing will appear, and I am not sure where to implement it. Here is the code to my archives page.
http://pastebin.com/Hb6bng9k

and here is the link to my page