The 25th Amendment states that the Vice President becomes the President if the Vice President and a majority of the Cabinet vote to remove the President from office. This means that the Vice President's vote is not included in the "majority". Secondly, Daniels is right when he says that Karen's vote is invalid, however it's not because Karen resigned. The National Security Advisor is not a Cabinet member and therefore her vote is invalid. Thus, the proper score would be 6 to 6 (not including Daniels' vote), it's a tie, therefore President Palmer remains in office.
Daniels: Tom! You know something, I'm beginning to think you're not on my side.
Lennox: Sir, I'm on the side of the United States of America and if you ever thought otherwise, you're very much mistaken.
Doyle: Turns out there was no mole. The terrorists were able to hack into our system because someone screwed up and logged the wrong security parameters. I'm pretty sure that someone was Milo and the proof is on his computer. I need someone to get it without him noticing.
Nadia: I can't do that.
Doyle: Why? Is it because you're too busy deciding if you want to sleep with him?