I'll reiterate some fo the coffeehouses that I have recommended in the past. Cafe Sperl - best apfelstrudel and the look that you are going for. Cafe Diglas on Wollzeile - really excellent cakes. Vollpension - run by grannies who bake their favorite cakes, fun for kids. Gerstner - an uber-elegant cafe (be sure to go the upper floors) with amazing sour cherry strudel. Alt Wien Kaffee - really cosy, local vibe and new to my list of favorites. Café Prückel - Breakfast at Tiffany's vibe, good for lunch.
I will also share cafes to avoid - Sacher, Demel and Central. These are all completely overtaken by Instagrammers and long lines. Sadly, they used to be great, but now they are tourist factories.
I will also share my new favorite with you via private message.
As an aside, you should know that when ordering cakes, you need to go to the cake display to make your selection. Viennese visit cafes in the middle of the afternoon.