Also worth noting is that you very nearly had it. From the log, we can't see what you put back on the deck from your Lookout on T12. But if, instead of gaining Gold with that first Governor, you had drawn 3 cards, you would have been able to buy a Province and win the game if you managed to draw the Gold that was otherwise in your next hand.
I disagree that Lookout would have been a good card for you; it helps your opponent's Watchtower/Hamlet/Peddler nicely, but doesn't contribute as much to your Governor-based deck. I think I like double-Hamlet on that turn; later on, a Governor for cards and pair of Hamlets can let you pick up two or three Peddlers plus another Governor or whatever.