Short of taking it somewhere for repair, it sounds like you're going to have to open up the case to get inside the drive. Once you've done that, you have two options:
1) If the connector is still functional you can connect it to your computer with the case removed and transfer your data, or
2) You can disconnect the bare drive from the guts in your case and connect it directly to your motherboard so that you can read the data from it.
....or purchase another enclosure of the correct size 2.5inch or 3.5inch