What I would do is try to repair the connector itself. It it's just a broken pin or something, there's a good chance that a bit of solder and fine wire could do it. The fix only has to last long enough for you to get the data off into the cloud, or onto another drive.
If you are not technically oriented, but still are comfortable tinkering with it, then buy another identical drive from Amazon or elsewhere, open the new enclosure, and swap the hard drive inside for your old one. Since you would probably need a new drive anyway, this doesn't really cost as much as it sounds.