Yes, MRPRichardson. Cooking is another equally valid translation of la cocina. And this sort of thing is the one really pet peeve I have about this site: they often only accept one translation, where the dictionary gives several. They score the answers as if translation was a strictly one to one process, rather than it being essentially a many to many process.
You're thinking about the personal "a", which does not apply to all direct objects, it applies only only when the direct object is an animated object (most commonly a person/pet). "kitchen" is not a person/pet, so it does not need the personal "a".
(Though I suppose if you really wanted to you could say that you were referring to the people working in the kitchen and not the kitchen itself, but Duo doesn't seem to want that in this sentence.)
Okay, I have a little question about english translation... Would it be ok to say; "I would have RAN the kitchen"?? ... Duo corrected me by saying " I would have RUN the kitchen"...
I'm not sure I understand the difference between the Run and Ran and why run is correct and ran is not!?
The past participle of "run" is "run". So it's actually "would have run", but it's a common mistake to say "have ran". http://www.usingenglish.com/reference/irregular-verbs/run.html
Doesn't like cooking for cocina. Wants kitchen. But go on Spanishdict.com and see that definition three from all three dictionary sources lists cooking as one translation. This is what I mean about why I became frustrated and drifted off this site a couple of years ago. That and I got beyond this for the most part.