London Hotel help

Is there such a thing as a hotel room with a washer and dryer? Maybe you need to look for an AirBnB apartment. But be aware that UK laundry dryers are not at all like those in N.America. Find out by a quick google.

I stayed in this flat in Kings Cross twice some years ago and would recommend it. Clean, well-lit with comfortable bed, washer/dryer and close to British Museum, Waitrose and several block from Russell Square station.

https://www.vrbo.com/103088

Will your trip be longer than 7 nights? If not, I'm not sure you'd really need to do laundry.

When looking for lodging it's best to start thinking in the currency of the country. $200 = about 153£. To me that figure might be a bit low for 2. I just looked at a random April date at the hotel where I usually stay and it was 190£ for 2.

Not sure if the American style chains like Holiday Inn/Marriott/Hilton etc have washers and dryers on site but you could look. Those will almost certainly be out of your stated price. I've stayed in several different hotels in London and none had washer/dryers on site.

There are extended stay type hotels that have self service laundries in the building but for one in your unit you will have to find either a serviced apartment or one via a service like Airbnb.

If you look on a tube map and figure out where zone 1 correlates on a regular map you should be fine. No more than 15-20 minutes to most sights on the tube.

Your budget is low for London. U may need to forego in building washer dryer for a laundrymat or drop off laundry close by.

Staybridge Suites have washers and dryers on premise and they recently opened a location near the Vauxhall tube

https://www.ihg.com/staybridge/hotels/us/en/london/lonva/hoteldetail

Finding a hotel with washer and dryer will be a challenge and will severely limit your options. There are launderettes that will do a ”service wash” (you drop off clothes in morning, they wash for you). If you are willing to do that, you will find many more options.

I can see why you are overwhelmed. You have laid out a criteria that is impossible to met. You need to establish a priority. I don't know how many hotels we have stayed in over the years in London. We have stayed in London at least month total. We have never had a hotel with a washer and dryer in the room let alone on-site. Frequently around the corner but not in the hotel. We have stayed in apartments with small washer and dryer but not ever one either. Don't think we have ever seen a king bed except the one time we stayed in the Intercontinental Hotel in London. Twin beds are common. Just ask for a twin room. Centrally located will push past your $200/night limit. Probably should be looking somewhere in zone 2. Outer edge of the metro area for London but convenient with the tube. En-suite is fairly common these days so that is not a problem. You may find a electric tea pot and/or coffee maker in your room but not much else to "make breakfast." You need to redo your list in a more realistic manner.

We've stayed at the Staybridge out in Stratford, which is not centrally located but has similar amenities as the Staybridge Vauxhall and were really happy with it- good breakfast included, on-site laundry room and king-sized bed.

The only King sized beds I have seen in Europe are in US chain hotels. For a washer, perhaps also a dryer, bit not always a sryer, an apartment is needed. Hotels do laundry for you for a fee and in the total price of your trip, it isn’t a major expense.
We loved staying in Chelsea. On other trips we have stayed near Kensington Palace/Hyde Park.