This is really going to be difficult. Once a hosting account is cancelled, content is essentially deleted/shredded. Retrieving it means going into "archives" which is a totally different function than backup/restore that is part of the hosting. All backup/restore functions are the site owner's responsibility and GoDaddy does give us those tools.
In addition, there are content ownership considerations. When you hit cancel, the dialogue prompts, "Are you sure...". Because you own your site's content, GoDaddy is obligated to remove it when you are no longer hosting with them.
I haven't had this happen in a long while (very early learning mistake!) so I'm not familiar with GoDaddy's current archiving tools/policies. So, you will definitely need to contact customer support/service ASAP.