Now, what would Drake do with the discovered continent? Return home with a boatload of pingwins to give them to Liz as a present? I'm afraid that she'd find it a lousy return on her investment and consequences for Drake could be really bad.
Colony in Antarctica is an interesting idea. What exactly the colonists would be doing? Even in the XIX, when it was finally discovered, it was so patently useless that nobody tried to colonize it.
I'm with the whaling-station idea. I don't know much about Antarctica (although one of my student buddies went with on a geological trip down there, and his only was that "its as cold as a witch's left tit"), so I'm not sure what type of soils (if any, from most of what I've seen on National Geographic the beaches seem to be rocky and the land covered in snow) are present where Drake would've made landfall, not to mention the growing season would be short wouldn't they? Plus, IIRC penguin meat tastes absolutely foul according to the sailors who had eaten it - oily, bitter and a couple of not so nice other terms were used to describe it. So, we might see a whaling station established - if they can grow anything there (did the Vikings grow stuff on Greenland?), but I'd imagine that it'd be lucky to make it through its first winter...