It was pretty clear from early on that she had a shardblade.
Thanks to the other posters, particularly Chosen, in this thread finding and inserting most of the important quotes.
I thought she probably killed her father with the shardblade, just because there was blood pooling around him doesn't mean a whole lot. Any subsequent blows after the initial death would have cut flesh, and she might have been flailing at him with it.
But then Chosen brought out this quote:
Ten heartbeats, to bring forth the fruit of her sin, the proceeds of her most horrific act. This it most easily interpreted as her taking the blade off her father's corpse. But then how did the Soulcaster get cut in half?
The Soulcaster looked identical to the one she and her brothers had found in the inside pocket of her father’s coat. -Page 95
Shallan fished in the safepouch inside her sleeve, bringing out her father’s broken Soulcaster. It had been sheared in two places: across one of the chains and through the setting that held one of the stones. She inspected it in the light, looking—not for the first time—for signs of that damage. The link in the chain had been replaced perfectly and the setting reforged equally well. Even knowing exactly where the cuts had been, she couldn’t find any flaw. - Page 575
I'm assuming that he was hiding it on himself, and the shardblade cut through it.
So how did this all go down? Nan Balat did something that upset their father during dinner. He started beating him viciously enough to break his leg horribly. He must have been about to kill Nan Balat when Shallan intervened and killed him instead. Did he summon the shardblade, then put it to the side, and then she picked it up and slashed him with it?