One way would be to download all the images to your computer and then use the bulk media uploader to put them into the new website. Trouble there is that you then need to reset them into your existing posts manually. If you have a large site, that would take a while.
Another option would be to leave them where they are, delete all the other stuff from the /site folder and forget about them. Your new site will continue to use those as the paths are already present in the existing posts. Any new posts should use the new path location for the new website in your public_html/ folder.
Another option would be to delete all the posts and media from the new site. Then run the Export script on the old site located at Tools -> Export. Then import that file (thus restoring all the posts you just deleted) into the new site, ensuring that you select the option to download and save all the attachments into the new site. Once that is done, go into the posts and check the image paths to ensure they are showing the new one.
Lastly, there may also be a code-based method such as in htaccess or a php script but that is going to be help beyond what I can offer.