The speaker wants to make it really clear that he hated her speech. "A new low" implies that it was the worst ever, and the adjective "bleak" is meant to intensify the sentiment.

It means that the listener found the speech very distasteful/repugnant. The phrase "to sink to a new low" very often refers to a moral judgement, and when it's impossible to sink any lower, you've hit "rock bottom". So for example, if a politician gave a speech about how to tackle illegal immigration and you found it to be inhumane/immoral, you could say that their speech, or that they themselves, "sank to a new low".

"Low" is a noun. "Bleak" and "new" are adjectives that modify "low". "Bleak" means dismal or discouraging. You could also turn "bleak" into an adverb: "She delivered a speech that bleakly sank to a new low".