In her aside, what do we learn about Gertrude and how she feels following her meeting with Hamlet in her bedchamber?

The scene involving Hamlet and Gertrude in her bedchamber (often called "the closet scene") is among the most pivotal in the play. However, Gertrude is not on stage alone following the scene—and she does not have an aside.
Still, we learn much from her actions as to how the scene has influenced her. In his final appeals to his mother in the scene, Hamlet makes requests of her. First, he suggests that she can combat her wickedness by way of resolve: "Refrain to-night, And that shall lend a kind of easiness To the next abstinence." In other words, she can be saved.
Hamlet puts before his mother a polemic: choose to align yourself with me [Hamlet] and justice, or align yourself with Claudius and wickedness. As a test, Hamlet asks her to keep his secret. He wants her to tell Claudius that he is mad, not "mad in craft." There has been much debate in the play about why Hamlet has been acting so strangely. Here, Hamlet tells her that he is faking his madness to shield himself from scrutiny as he investigates his father's murder. If she wants to do the right thing, she needs to keep Hamlet's secret.
In the scene immediately following, she does just this. She sees Claudius and tells him that Hamlet is mad. By doing this, Gertrude begins to break the bond between herself and Claudius, thus redeeming her character.