There are some things you can do. But keep in mind that if you open the plastic case, you will void any warranty that you currently have on the drive.
The first thing I would do would be to create a Linux Live DVD, and boot the computer with it. I would then try to read the drive from within Linux Live. Sometimes Linux can do a better job of reading a drive than Windows can. Of course, you will need to have a drive to copy your files to. After you finish, shut the computer off, remove the DVD, and turn the computer on, to get back into Windows.
To create a Linux Live DVD, go to any of the Linux websites and download one of the current .ISO files. Put a blank DVD into the drive, right click on the .ISO file that you downloaded, and tell it you want to create a DVD from it. The process is pretty simple and straightforward. You could go to https://linuxmint.com/download.php and choose any of the eight files. If you aren't sure which one to choose, select the 32-bit XFCE file.
If you can't get to your files via Linux Live, then you could try to take the case apart to get to the actual hard drive. Once you have removed the actual drive from the plastic case, try installing it in the computer as an internal hard drive. You may be able to read it directly in Windows when you have done this. If not, then reboot into Linux Live again and try to read the drive.
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