The speaker is telling Anna that because she is here, he knows that they are both never going to accomplish as much as they could.
firmly establishes = proves
what I've long suspected = something that I have thought for a long time
desperate = it can mean "in great need," or "hopeless," but in some cases, such as this one, it can mean "unable to change." The adjective "hopeless" can be used the same way. For example, in the phrase "hopeless romantic" it means someone who will never become more practical and stop being so romantic.
"underachiever" = Someone who chooses to accomplish less than they are capable of. (Lazy is USUALLY, a pretty good synonym, but not always.)
So, a "desperate underachiever" is someone who doesn't want to accomplish very much, and will never become more motivated.