How should we go about finding a place to stay for a few months in London?

Your budget is pretty big. I'd find a fancy place on AirBNB and negotiate a three month term. The standard responses of GumTree and SpareRoom don't apply because of your budget.

When I first arrived in London I stayed in a Serviced Apartment / Business Apartment for a month or so. I've also had various friends come over for work who found fully furnished / equipped apartments for terms of 1 month to 6 months. I stayed in a place with City Apartments http://cityaparts.com/ but there are loads of other services http://www.apartmentservice.com/ I would use search terms such as executive apartments, serviced apartments, short term rentals, these sort of places are more expensive than a renting a flat but they are also usually very central London and they have all the cutlery and microwaves, WIFI, and stuff so you don't need to bring anything but can cook and live a normal life - rather than living out of a hotel for 3 months. Your budget will be more than enough to afford one of these types of places.

Many estate agents also list short term rentals -- for instance, I remember that Foxtons has a specific checkbox for them in the search. Your budget is easily large enough to rent http://www.foxtons.co.uk/search?submit_type=search&tag=&search_form=&lat=&lon=&radius=&search_type=SH&center_point_name=&keyword_value=Victoria+station&price_from=&price_to=750&bedrooms=1&bedrooms_max=&prop_type=flats.

Unless you want to walk, you do not need to be near Victoria. For example: Brixton is nowhere near Victoria. It is 9 minutes directly on the tube. Earls Court is nowhere near Victoria. It is 10 minutes away on the tube. Similarly, Marylebone is also central and easy to get to. http://www.london-tubemap.com/journey_times.php is a great primer so you can get a sense of how far, in real terms, how far away places are from one another. Treat an interchange on the tube as 2-3 minutes. You can also use the http://journeyplanner.tfl.gov.uk/user/XSLT_TRIP_REQUEST2?language=en. Your budget is pretty good for a rental. The terms to use are "short term let" or, if you want one, a "serviced apartment." If it were me, to save the faff of having to get utilities in my name, sort out a TV licence and so forth, I'd go with a serviced apartment. You will find nice central places to be for 3 months in areas like Belgravia, Chelsea, Pimlico, Vauxhall, Bayswater, Marylebone, Bloomsbury, Farringdon. I'd also consider - if only for comparison - looking a little further out of the centre but still near a tube line at some of the leafier areas (so: to west, somewhere like Notting Hill or Maida Vale, south somewhere like Clapham or Battersea, north somewhere like Belsize Park, Primrose Hill or Hampstead). You'll get your space, you'll still be close to everything you want, but your budget will stretch a little further. London is a sea of little villages, so while learned people will give you sage advice about living in a particular area, there are lots of lots of nice places to live. That said, South London is poorly served by the tube. If you want to get a sense of what longer term rental prices look like, then http://www.rightmove.co.uk/property-to-rent.html is a useful place to start. You can get a sense of London postcodes http://www.londontown.com/LondonPC/. WC1 and WC2 are the centre of town. EC1,2,3,4 are the City (i.e. financial district) and edges thereof.

My family has done a serviced apartment (for shorter stays-- a week or so?) in Kensington. Really great location, ability to cook meals in-house, all furnished/with all the needed bedding and so forth. It was more expensive than a normal apartment or most AirBnB type options, but the convenience of having everything supplied right there was very useful.

MuffinMan has wise advice - your budget suggests that you aren't expecting to have trouble covering your costs while here, but you will if renting need to pay for council tax, utilities, TV license etc. which will drive up your costs. Also, VAT is 20% here. If you go out the centre, for that price, you can stay somewhere bigger with nice parks and cafes. Victoria and Marleybone are both central areas and unlikely to be quiet at weekends (I work near the City in Bloomsbury/Farringdon and everything shuts here at the weekend - Marylebone is more touristy).

The people who own the flat next to our shop rented it out during the Olympics through http://www.onefinestay.com/london/central-london/, which seemed to work out well on both sides.

We rented a flat in London for 6 weeks last year through http://www.ivylettings.com/, and had a very smooth experience. Lots of pictures and details about the locations on the website, and the flat we ended up in met our expectations completely. We did the entire transaction over their website/email/bank transfer. The price we paid was all-inclusive, so no worries about tax, utilities, TV license etc. My 2 cents on where to stay--if the budget is being provided to you by her employer (e.g., it isn't *your* money exactly) then stay in the nicest/closest to town place you can (and within short walking distance to Tube and or busy bus line). The closer you are to the center of things, the more you'll be able to do more conveniently/easily. Have a great time!

MuffinMan makes good points about London geography and getting around, but your quality of life will greatly improve if you can avoid taking public transport at rush hour. Space will be easy based on your budget and London is surprisingly easy to find quiet. The difficulty is that is usually requires going to visit a place to know this, as it can vary greatly between neighbouring streets. For me, nearby green space is critical so I live by a park, but perhaps you'd value being closer to the West End where much of the evening activities take place. Don't worry about internet, it's cheap and fast here compared to SF. As an aside, I'm not sure what preparations your wife's company is making, but if you can organise a UK bank account before you arrive, it will make things much easier. One way to do that would be to go speak to HSBC in SF and see if you can get a Premier account. HSBC is everywhere in the UK and they'll use your US credit standing to get something set up.