Short lets in the UK?

[BabCNesbitt]

Posted this in the Living Overseas forum, but wondered if I might have more luck here.

DH and I moved to NYC with our daughter last March, and we'll be moving to Colorado permanently this summer. But DH has a temporary position in London from May to July. Yay! of course, because it'll be lovely to spend time with friends (and go to free museums instead of paying $20 a head, cough). But we have to find our own accommodation (no friends I want to impose on for 3 months with an 18-month-old) and I'm having trouble finding an affordable place to stay for that time. Gumtree is pretty useless - it's mostly rooms on there, and it's impossible to know how legit they are - and all the other short-term lets I've seen seem to be aimed mostly at businessmen whose companies are paying for their accommodation. I've tried AirBnB, but again, there's nothing suitable in our price range (up to about 1600 quid a month, I guess). There are lots of studio flats, but those aren't great when you want to get a toddler to stay asleep after 7pm!

Has anyone else had any luck finding a short-term let in the UK? Is there any point in talking to estate agents? I know there are cottage letting sites, but there aren't that many quaint cottages for under 1600 a month in London...