I'm not sure of any companies that do that kind of stuff, but if you wanted to try it yourself us here on the board could surely help you. Based on the size of the site and what I saw it should not take you very long at all to move over.
First thing I would do is ask the new webhost if they are able to do a "Migration" if say the previous host and this host used the same control panel software they could just automatically transfer the data over, or I know cPanel, will transfer accounts from plesk,helm and some others. Check on that you would just need to give the new host your login info they would enter a command and site moved! Thats if they are nice enough to do that for you, or if they are setup to do that. But its always worth a try