@Matt Joplin
First, you need to realize that the state of Washington (and lots of cities/counties) are cracking down on unpermitted additions and ADUs. Mainly because the STR issues and that the rising home prices are to be blamed on STRs. So you have a lot stacked against you to begin with. Regardless if you are in the right or not.
A couple things that need to be cleared up for you. Your "kitchenette" is a full blown second kitchen. Don't even both trying to call it a kitchenette to try and get around codes. Now the IRC doesnt put a restriction on second kitchens but it does put a restriction on number of units in a house. When you have a bedroom, bathroom & kitchen in a separate part of the house then you are effectively creating a second living unit. Now, is it truely separate? More on that later.
Next, by state law, ADUs can be attached or detached. This trumps the city code. You can try arguing this but I'll get to that later.
Next, just because it was existing when you bought it, does not mean it is legal or that it will get grandfathered in. If it wasn't properly permitted it is now your responsibility to fix it to meet code.
Next, state law currently requires that an ADU and main home be owner occupied to be able to be rented out to a full time tenant. That means you can only rent one of the units. If you dont live there you can only rent out the main house and not the ADU. It is also currently illegal to rent an ADU as a STR without being owner-occupied (this varies by city/county so check yours first). There are some proposed laws that would change the owner-occ requirement for LTR but dont bet on the STR restriction changing.
I see you have already tried to dialog with the city and they are basically all against you. So odds are that if you try to keep going in yourself you wont get anywhere. To be frank, you are not a professional that has experience with building or building codes/zoning (the vast majority of people arent) so even if you present codes correctly they arent going to take you seriously. You are going to need to get a professional to help you with you case. I would actually go to an architect first, over a lawyer. Not only are they going to have expert opinions on the local codes, they can help navigate with the city and help this get worked out. If you go in with a lawyer they will get defensive right off the bat. Save the lawyer for when you actually have to take them to court. As I see it here are your options (with an architect first):
1 - You can try to present your case that the upstairs second unit (what they are calling a second unit) is not separated from the rest of the house and is just an extension of the main. That there is no intentions of and no current way to separate the second unit. BUT the fact that you have already rented this out (or were wanting to) proves that this is, and was your intention to, treat this as a second unit. This will be hard pressed to pass muster.
2 - You can try to make your home a duplex. I dont know the layout of your home but 'usually' converting to a duplex is not that difficult. Your hurdle is the lot size requirement. This would require a variant though. It is possible but would take time and effort and could still be rejected. I think your best bet here is to prove that the addition/remodel was permitted. This would stop them from requiring you to remove the kitchen but there would still be restrictions on renting it (either STR or LT). But as soon as they cant make you remove it, I would think, they would work with you to make it compliant.
3 - Making it a legal ADU. First, this would not require much work. You do not have to have separate utility connections and meters for an ADU, unless the city specifically requires it by code...most dont. Duplexes aren't required to have separate connections and meters...and traditionally they havent. Again, the city could require it but that doesnt make much sense to me. This does come with the rent restrictions that I mentioned above. But it does give it extra resale value. I would at least do this so if you cant do what you wanted to you can at least sell it and not have any issues coming back to you later.
4 - Remove the second kitchen and turn it into a true kitchenette or wet bar. Now, removing it all would be the easiest route with the city. Labeling it a kitchenette or wet bar is another route, and legally you could do it, but odds are you will have a hell of a time getting the city to buy off. Even with an architect on board. I've just seen a lot of push back, even to this, even though they have to legal standing. The cities get to do as they please until they get sued. This would at least allow you to rent out the entire house as either a STR or LTR.
5 - Getting a lawyer and suing the city. This is obviously a last resort and a very expensive option. The reason cities have become so authoritarian is because they know the average person doesnt have the money to fight them in court. You will more than likely have to go through an architect first and have your options, that have been considered legal and viable options, that meet the code and presented by a profession still get rejected before a lawyer is going to have the ability to see that you have a reasonable case against the city. There might be other requirements before going to court too....like going through several appeals processes. Which could take a very very very long time to go through.