Okay, so I decided to test it again, with the exception of not doing anything else that could be using the same render engine. I restarted my computer, logged in, and decided to run CCleaner a few minutes after logging in. I left all of the settings still in tact like before, not changing any of them. So, I clicked Run Cleaner and it successfully cleaned IE's Temporary Files, History, and Cookies. After cleaning, I closed CCleaner and opened IE and used it for a few minutes and eventually closed it. I opened Task Manager briefly to see if the process was gone and it was. So, I went back to CCleaner and did the same thing like before. Surprisingly, the prompt didn't show up at all and successfully cleaned IE's Temporary Files, History, and Cookies. Lastly, I did the steps one more time just in case anything different happened. After I opened IE again for a few minutes and closed it, I went back to use Task Manager to see if the process was gone. However, the process was still there and I waited to see if it would disappear, which it did after a couple of seconds. After I made sure IE's process was gone, I ran CCleaner, clicked Run Cleaner again, and successfully cleaned IE's Temporary Files, History, and Cookies yet again.
So, after doing that I don't think it's necessarily an issue with CCleaner itself, but rather Internet Explorer. It could be possible that IE isn't closing immediately, or otherwise taking longer than usual. I can check to see if the version of IE on my computer needs to be updated if that would solve anything. Thus, it seems like the issue is resolved for now. I thank you for the support and if the same (or a different) problem arises, I'll keep you updated with that.