It's good that you have some criteria in place. Now you will have to prioratize that criteria.
I would start by getting a tube map:
http://www.tfl.gov.uk/assets/downloads/standard-tube-map.pdfIt shows stations that are wheel chair accessible.
You should also look for rail maps for lines that go into Liverpool Street Station or St. Pancras if you decide that easy access to that station is more important.
I would also get a map of London so that you can see what you're dealing with. Just to give you some perspective, London is about 659 square miles.
You might want to start by looking at stations that are on lines that go into Liverpool Street directly. Your budget is pretty tight for what you want so I would start at Zone 3 and beyond. The farther you get away from Central London, the more you get for your money.
Here are some property websites:
www.rightmove.co.ukwww.primelocation.comwww.findaproperty.comThis website will give you an idea of neighborhoods:
www.upmystreet.comFurnished flats are pretty normal and just make sure you tell your estate agent exactly what you need regarding bedrooms, no stairs, lift etc.
Estate agents are very area specific.
The best way to find out if a neighborhood has everything for you and that you feel comfortable is to spend some time there just walking around. A neighborhood that might be great for me, might feel off to you. It's very subjective and as I pointed out before, London is huge. There is something for everyone, you just have to do your research which if probably going to involve a lot of legwork.
Good luck!