There are at least four ways through which you should be able to delete a video from your phone:
1. If the video was taken using Lumia Camera or Microsoft Camera and is your Camera Roll, you should be able to tap an hold on the icon of the video until the "delete" icon appears. (It's still worth another shot even if your description says that this method doesn't seem to work.)
2. If the video is as described in #1, you can also use the gallery viewer in either Lumia Camera or Microsoft Camera to delete the video file.
3. Use the official Microsoft Files App. This app can be used to delete almost any file that is in your phone wherever it is saved (except system files). The app can be downloaded from the store for free.
4. Perform a hard reset - but of course this will delete everything from your phone (except the system files). Most people will not consider this an option due to the implied inconvenience of restoring all data back, but if the videos are extremely sensitive, this option should still be a strong candidate.
If these methods didn't work (except #4), we may need screenshots in order to provide more concrete "assistance". You would also need to create a Windows Central account to respond to this thread.