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There are so many incorrect tinfoil hat solutions here. Allow me to present a correct tinfoil hat solution that fits into your "without formatting" requirements.

Even magnetic force microscopy isn't going to get the files back if they've been deleted properly. Burning the drive is an extreme tinfoil hat option that is completely unnecessary.

Taking Your Tinfoil Hattery to the Next Level

What are the steps I have to take if I want to completely erase all traces of these files without reformatting or doing something similar?

1. Disable Windows Hibernate
2. Disable Shadow Copies
3. Delete the information in the AppData icon db
4. Move the temporary files to another folder, FolderA, along with recent items and shortcuts.
5. Move the files in question to FolderA.
6. Rename all the files in question to something random. Private Health Information.csv? Secret Chinese Dissident List.pdf? You shredded them, right? Yeah, the file names are still searchable, even if nothing can be recovered, even after shredding them, in a large majority of cases. In many countries, the mere possession of banned materials will get you arrested and/or executed. Religious documents, anti-government propaganda, etc. Just finding a file name is all they need to compel you to donate your organs. Use a tool like Recuva to verify this. Even if it's securely deleted, having the file name listed would make people realize you've been in possession of those documents. Make it something like "derp.exe."
7. Securely overwrite every single renamed file.
8. Delete these securely overwritten files.
9. Reboot in safe mode with a command prompt, or use a separate drive to boot off of. I recommend a separate drive.
10. cipher /w:drive letter 2 times for good measure. One should be enough, but if you have an SSD**, you will need 2 or 3. This completely defeats wear-leveling as every single block of space on your drives has been filled up with garbage.
11. Unplug the power for 20 minutes. Start the computer, and use normally.

Keep in mind that if your drive/files have already been copied by someone else before the deletion begins, you're boned (network transfer of hard drive data / physical copying) if they find your Chinese Dissident List.pdf. A simple diff of the contents of the drive will reveal it, and you'll end up generously donating organs to the local cadres.

Solid State Drives

** Now, let's talk about SSDs a little bit more. You can get by wear-leveling using cipher a few times, but it's honestly really bad. You will likely hurt the life of the drive by doing this, if not outright kill many blocks. This is bad advice for an SSD, but I'm including it because it is possible. It just may come with undesirable side effects.

Alternatives

Would your life be in danger because of this? Alternatively, encrypt the drives with a super-long password, then format, and dban boot and nuke, then reformat.

Otherwise, shred the drive.