The sentence first contains reported speech (in Konjunktiv), and then continues in indicative mood as if talking about facts. The whole sentence is a mix of Mrs. Dursley's and the narrator's truth, and some artistic license is used in the grammar.
Mrs. Dursley behauptete sogar, dass sie gar keine Schwester hätte.
The main clause is indicative mood ("behauptete").
The dass clause is reported speech in irrealis (i.e. contrary to fact). The Konjunktiv II "hätte" is used because, despite Mrs. Dursley's assertion, we know that she does have a sister.
Denn diese und deren Nichtsnutz von einem Mann waren so undursleyhaft, wie man es sich nur denken konnte.
This is written in indicative mood again even though it's clear from context that at least "deren Nichtsnutz von einem Mann" is Mrs. Dursley's perspective, not that of the narrator.
JK Rowling likes to mix this up in English, too, it's not only in the German translation.
"Mr. and Mrs. Dursley of number four, Privet Drive, were proud to say that they were perfectly normal, thank you very much."