I have to break down this speech and define one's "likely experience." My problem is that, like with most Shakespeare, once you break it down, what made sense when you first heard or read it, is absolute gibberish. "Babbling gossip?" An interesting notion, but ironic since Viola is in disguise and can't babble or gossip about anything.
Any ideas on how I can structure a thesis based on the prompt and my brief experience of the speech as demonstrably incoherent?
Nonsense is humor, in Shakespeare's world! All kidding aside (I think Shakespeare inspires that in me), you have to look for the deeper meaning here. In Shakespeare, it is all about subtext and double meaning. Look for that and you'll find the humor.
You might want to think about the way in which Viola could be parodying the excessive love displayed by Orsino, who famously was denoted as being more in love with the feeling of being in love than with Olivia. Viola of course finds herself in a very tricky situation: she needs to woo Olivia for the man that she loves, and so perhaps we find in this famous speech her own "take" and parody of Orsino's love-sickness.
Keep in mind that with most of Shakespeare's comedies, the humor comes from mistaken identities, dreams, magic, people in disguise, twins, impediments to love, practical jokes, and sometimes, a bit with a dog. There are also many subplots which add to the complexity and the likelihood of confusion and laughter. It is better in this play, since Malvolio, the villain (just look at his name to know), gets his just deserts for his arrogance and ego.
Viola is the only character aside from Feste who is able to comment on the action as a whole in the play. From her spot in her disguise, she is able to take full advantage of double entendre and the practical jokes that abound in the action. She does so with gusto, and thoroughly enjoys her role.
Take another look at the speech with this in mind, and see what else you can get from it.
Yes, I can type "12th," just not on this new keyboard, quite yet! :)