I think you're referring to Edith Wilson when Woodrow Wilson was convalescing from his stroke--which, by the way, has NEVER been proven.
Anyway, when Garfield lay dying, the Country was run by... no one. It was the Summer, Congress wasn't in Session, and there really wasn't a whole lot to do.
Unlike today, when the Constution states the Vice-President shall Act as President if the President is ILL, there was no such language in the Constitution then.
Then Vice-President Chester Alan Arthur was the EXACT OPPOSITE of John Tyler--he followed the Constitution to the LETTER and assumed no Presidential duties until Garfield was declared dead.
"I say YOU'RE the CUTEST one. No, I say YOU'RE the CUTEST One. And we go on like that from dawn to three."