This question pops up a lot.
The difference between GA and WA is that GA is the subject marker, while WA is the topic marker.
GA follows the subject of the sentence, which WA is then used to provide information or qualify a subject that has already been identified.
Consider: "O sake o nonda koto GA..." the subject of Sayaka's question is simply being identified as "the action of having drunk sake".
In Kenny's response, he is now qualifying the previously identified "action of having drunk". He qualifies the subject identified as one that "doesn't exist" (ie. arimasen".
GA identifies a subject in a sentence. When wanting to provide further information about the subject, WA is used.
Another, easier way to think about it is in terms of emphasis that the two particles place. GA emphasizes the subject before it, while WA emphasizes the information after it (though WA is still a particle and hence is immediately after the topic of the sentence).
For example, if someone asks me "Anata WA dare desu ka?" I would respsond "Watashi WA Pascal desu". In both cases the WA proceeds the emphasis of the sentence. In the question it is "Who?" and in my response it is "Pascal".
Another case may be "Pascal GA dare desu ka?". In this case GA is used to mark "Pascal" before it. The emphasis is on who PASCAL is. My response would be "Watashi GA Pascal desu", since I'm emphasizing that *I* am Pascal.
In the case of Kenny's brother, I think Kenny could say "Watashi WA o sake o nonda koto GA arimasen GA ani-san WA mainichi nomimasu."
The two WAs here I'm actually using to indicate contrasting terms, which is a function of the WA particle. The first GA serves the same function as in Sayaka's question, ie. to identify the subject before it/ place emphasis on the "act of having drunk". The second GA indicates a contrasting clause and has the effect of saying "but". You don't generally find Demo in the middle of sentences.
I think my explanation is right. My Japanese is a bit rusty right now, so any corrections are welcome.
Lesson 3.2, Wa and Ga