I know I can say that an apartment is furnished or not, but can I use the word
furnished with a kitchen?
I am trying to describe a kitchen that is equipped with the necessary tools for cooking and refrigeration.
Yes, you can. "Furnished" means "to supply (a house, room, etc.) with necessary furniture, carpets, appliances, etc." and a kitchen is a room, so it would fit.
furnish is usually used to mean "provide with furniture," it can also refer to other needed items. When I say:
I fully furnished my brand new apartment over the weekend
Few people would suspect that I mean that I only supplied a brand new apartment with furniture, without other appliances.
However, when talking about specific rooms that have a special designation, then "furnished a kitchen" or "furnished a bathroom" can sound quite odd, though not technically incorrect.
In the case of a kitchen I'd say:
I equipped the kitchen / I got the kitchen equipped
I bought all the necessary appliances for our kitchen
Usually you would say:
In the context of apartment rentals, this would mean that the apartment has a stove and a refrigerator.
That said, such a short phrase doesn't necessarily eliminate all ambiguity, as is discussed on this website.
Getting back to your question, one could also say
fully furnished kitchen
I would understand this to mean that the kitchen has all appliances, even a microwave.