I'd give the $10 USB Audio Adapter a try, as I believe you have exhausted your options given that you have done everything including reinstalling the correct audio drivers as Patio suggested and you followed up with already tried that. You have also stated that its not muted. And all other details would otherwise suggest that the audio adapter is blown. I am 99.99999% certain that this USB Audio device will be a solution to your problem. Its a band aid, but you'd have your sound back, and the quality of the sound is equal to what you had before probably unless you had a surround sound setup.
Only method of digging further is if you happen to have an oscilloscope handy and are familiar with digital/audio circuitry, and component identification to see if the audio signal is say making it to the amplifier chip and no output from amplifier chip to find failure at component level vs device level that we are currently at. Guessing this is beyond your means of troubleshooting? Also its possible that the realtek chip could be an all in 1 chip where you cant gain access to an audio input leg to the amplifier before the output leg of the amplifier to test for audio signal generated, but with blown amp output.
http://www.realtek.cz/realtek-datasheet.php Most people go with the band aid vs troubleshooting to the component level unless you are fixing say a rare obsolete piece of electronics like a rare arcade with a blown audio output and there is no adapter replacement option available other than fixing it at the component level which I have done before myself. When it comes to PC's it simply is not worth that much work to get audio back, especially when it can be resolved inexpensively!
Below is an example of pinout that can be acquired if you know which specific chip your system is using. Looks like its an all in 1 package.
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