Hi, F. one. We go to England every year to see my husband's family. I believe you're married to a Brit as well, right?
I will have to check out the NH hotel chain someone mentioned! But I have not ever seen or heard of them being in London, so go online and find out. AAA would be another good place to start. If you are not a member it is well worth joining -- they are one of the few real travel agencies left in existence.
Are you IN London the whole time? If so, be prepared for evil sticker shock. You probably already know this but with the exchange rate, London is hideously expensive for accommodation. Here are some ideas:
For what you want (microwave, mini fridge, etc.) you have to ask, ask, ask. Things vary so widely. Check individually with various B&Bs or small hotels -- things that call themselves "hotels" often are more like boardinghouses, with very small rooms crammed into buildings near major rail stations (Paddington is one area with tons of these little "tourist hotels.") Use TripAdvisor to check them out; generally I find the reviews pretty accurate but you have to take things with a grain of salt online. Some of these small, independent hotels are nice and clean; others are run-down; even the nicer ones may have carpets that seem old and worn, bedspreads that are not great to look at or sit on (but are clean!), old furniture, and so on. Standards can be very different in these hotels.
Always ask specifically about the things you want like a microwave and fridge, and definitely ask whether rooms are "en suite" (meaning bathroom IN the room) because some hotels and B&Bs still do not have toilets and showers in the room but down a hallway for shared use!
Also be sure to ask if young children are welcome. We have found that some of the nicer small hotels and B&Bs do not permit young kids.
Chains: Yes, they usually have a small fridge but not a microwave -- you may have to ask for one to be put into your room specially. You won't find K-cup coffee pots in rooms, but you will get an electric kettle, cups, and tea and instant coffees etc. most everywhere including small B&Bs.
You can find Best Westerns in London and they are fairly reasonable by London standards -- "by London standards" means you might pay $200 to $250 a night if you're really lucky. There is also a chain we like a lot called Ibis (one near Euston station is the main London one, another is at Greenwich) that is nice, clean, basic, small but very good rooms, has amenities. All the other big chains are there (Holiday Inn, Hiltons, Hampton Inn) but pricey, though Hampton can be a good deal. It's key to search hard online for deals! We got a Hampton for a great rate, I think around $200 a night, but that was in York, not London. Look for special rates and grab them online. Our Hampton had a mini-fridge and teakettle etc.
Someone mentioned a caravan...not in central London....
The farther out you stay, of course, the cheaper things will be. You can get better rates staying far out on the tube or train lines, so if you are willing to travel into the city each day, definitely investigate that. Get the most recent Greater London map to help you as you search or you could end up who knows where if you book someplace but aren't quite sure where it's located relative to a tube station or rail station. The Greenwich area is easily accessible from the city center by train. Does your husband have relatives in the London area who can help you pinpoint areas?
I don't know how to start with finding a flat for such a short term but I'd ask AAA -- some folks letting flats may not be reputable. Others may not want a young child. A real estate company that only does "holiday flats" would be a good bet, rather than going with individuals online whom you can't check out.
One tip: Remember that in the UK there may not be heating as efficient as we have here and there may not be elevators in older buildings including residential buildings. Just something to keep in mind.
One other tip: One way to save a lot of dough is to find the nearest grocery stores (in London these often are small versions of large chains -- look for Tesco Express; Marks and Spencer; Sainsbury's; Waitrose) -- and get food that you eat in your room for breakfast and as a picnic at lunch and then have dinner out. It's even cheaper to have your main restaurant meal at lunch and have a picnic dinner in your room. We do this a lot to save money on London's very expensive food.
Have a blast. Remember the saying from Samuel Johnson: "When a man is tired of London, he is tired of life." Best place on Earth. But take your MOST comfortable walking shoes!