I’ll give you some advice for recovering and destroying the appropriate files, but I don’t think you’re going to like it. It may not work, and if it does, you’ll have to pay a lot of money for it.
But first, let me point out that you got yourself into this situation. If you had backed up all of your files, and encrypted the sensitive ones, all you’d have to do now is buy and install a new hard drive.
Backup is essential. Simply put, you should never have only one copy of anything, and those extra copies should not be inside your computer. See How Do I Backup? for some practical advice.
In addition to backing up, you need to encrypt any file that you wouldn’t want another person to see. There are plenty of good programs for this job, but I recommend the free, open-source TrueCrypt. It can encrypt your entire drive, but I recommend leaving the drive alone and creating a TrueCrypt volume. When closed, this volume is an unreadable file. When you open it with your password, Windows sees it as another hard drive.
Okay, enough of the lecture. Now onto solving your problem.
There are services–not the cloud kind–that can take apart your broken hard drive in a clean room and recover the data (or at least try). These services aren’t cheap. Assuming they can recover your files, you’ll have to spend hundreds, or possibly thousands, of dollars.
Because I’ve never had the need for these services, and I haven’t found a practical way to test the quality of their work, I can’t really recommend one over another. So I’ll just say that I’ve heard mostly positive reports about the two best-known recovery services, Kroll Ontrack and Drive Savers, while adding that lesser-known services may be just as good.
And what about those files you need destroyed? Although Ontrack promises to keep your files confidential, only Drive Savers offers a special High Security Service. Your files are closely tracked, the media kept in a safe, and the whole process follows government protocol.
By the way, if you don’t care about recovering files and only want to make the drive unreadable, you can physically destroy it by drilling multiple holes all the way through the drive, or by smashing it many times with a heavy hammer. While this is probably the most fun way to destroy a drive, it can also be dangerous. Wear goggles and protective covering over your nose and mouth.