Looking for a 2 or 3 bedroom apartment near Borough Market London...

http://londonist.com/2014/12/markets-in-london-mapped

Why do you think the area around Borough Market is your husband's type of area? Clarity.

Personally I find Portobello Road, Camden Lock, Spitalfields and Brick Lane markets enjoyable to explore and meander.

I know of several to the east of Borough Market, down near Tower Bridge. However, they all have tons of stairs, and I know you don't want that.

Look at the other market areas Claudia has given you. Then search those areas.

You may want to search some of the Premier Inn hotels. They are really great, and budget friendly. They have connecting rooms and family rooms; perfect for your group. And no stairs, plus a lift (elevator) for the senior in your group.

There is one near Borough Market:
http://www.premierinn.com/gb/en/hotels/england/greater-london/london/london-southwark-borough-market.html?cid=GLBC_SOUANC
Now showing Price from £136.50.

I think you are going to have to consider hotels in the mix, as you are not having a lot of luck finding the apartments you'd like.

No info for you on where to stay, I'm afraid. But concur that all the markets offer something for all. Borough did stand out for the foodies in my group though. Since it will be Nov, you may want to consider being on other side of the Thames from Southwark (it as limited MUST-SEE sights) Not a huge issue but if weather or tired feet are an issue, maybe consider it.
Oh my! Just realized you are party of 10? You may just need to take what you can get. With the 90 day requirement rentals are not as numerous as once were.

No it's 10 nights. Sorry we are just 3 people. My hubby is a real foodie type of guy. He is also mobility challenged so no stairs would be great.

Fixed my original post to say 3 people 10 nights