It's a pain sorting something like this out and I've struggled with it on occasions too. You've tried the right things but it may be that you have a driver issue. You are running W10 but haven't said whether you upgraded from an earlier OS. Did you have speakers or headphones attached and working previously? I'd try plugging in and unplugging the headphones a couple of times and be sure to check that you are using the right socket. On my tower the colour code for microphone and speakers is similar and easy to confuse.
See if you can find within Control Panel, Device Manager and make a note of the details of the sound card. Does it say the device is working normally? Try updating the audio drivers. You could visit the manufacturers site. Restart the computer and Windows should detect your speakers/headphones . Still no luck, then I'd uninstall the audio drivers restart the computer and see if the default drivers that Windows 10 installs work. With my desktop there is a separate entry under Control Panel for Realtek HD Audio Manager besides the drivers in Device Manager. You're working towards a big green tick on the playback device set as default and enabled in your screen shot above.