"What a career!"
In exclamations we normally use vilken.
The construction vad för karriär is used a lot, but it's a question, to specify what career we're talking about. There usually won't be an article in it, although there is at least one set expression Vad är du för en? (≈ 'Who are you?') where it's used.
Thank you very much -- I guess I will have to remember it as "Welch eine Karriere" (also a turn available in German). Alltogether, all of those seem to have direct analogues in German:
"Was für Karriere?" -- "what career?"
"Was bist Du [/denn] für einer?" -- colloquially something like "Jeez, what kind of [person/idiot] are you?"
(somewhat to the point of "what kind of institution did they let you out of?")
I may be a bit delayed naybe, by now, you have already managed to remember this formulation, but as I see that you also learn Spanish, you could also derive this formulation for this language. In Spanish, you would most likely say »¡Qué carrera (profesional)!« So, it may be idiomatic for these two at least. Whether it would also work for French, I am not sure, but it most likely is the case here as well. »Que carrière !« or so, I am not sure about the word's gender in French.
I'm not sure if I have any good answer other than that you don't use indefinite articles after "vilken" or "vilket". The difference between "What career?" and "What a career!". Is just in that the first is a question.
"Vilken karriär?" - "What career?"
"Vilken karriär!" - "What a career!"
No, none of the other question words can be used in statements like that.
Also mostly the sentences are longer which makes it easier to determine if it is a question or a statement without having to observe the change in intonation.
"Vilken fin bil du har." - "What a nice car you have."
"Vilken bil är finast?" - "Which car is the nicest?"
"Vilken karriär hon har gjort!" - "What a career she has made." (I'm not sure if you would say so in English the essence in the statement is that she made a really good career.)
"Vilken karriär har hon valt?" - "What career has she chosen?"
Real life usage can sometimes be blurry, but the words do mean different things in English and their English meanings correspond closely to those of their Swedish counterparts, so it's not helpful to treat them as synonyms in a language course. Here's a link to one site that tries to explain the difference: