Wells should have asked to see the President right then and there.

The President has complete control to deal with the situation even if there are hostages and risk to it. If Wells was straight-forward with Shore and demanded to see the President that she had information regarding a potential connection between the Deputy-Director of the CIA strangely confessing to the murder of a possible terrorist, the bombing of the Capitol, and the unique survival of the man the President was supposedly going to tap to be the next Vice-President; Kirkman would have absolutely heard her out. She literally had a text from an anonymous number telling her not to say anything to the White House Chief of Staff on her phone. Showing that to Aaron alone would have been cause enough to immediately alert the President. Additionally the President and the many government and law enforcement institutions are completely capable to dealing with the situation. She could have walked out of the White House and pretended she didn't say anything to anyone while internally the government looked into it. That was a stupid move on her behalf.