In that case due to the resolution I would probably go for a second hand card, such as those listed on the linked Akasa graphics cooler page. You could look in the members market here and see what there is amongst that selection, then buy a cooler like I've suggested. You would then get a quiet card that can game adequately well for that resolution, while staying roughly in budget. You never know, you may even find a passive card on sale anyway.
If you don't wish to do that then from what's available brand new here, I can't see you doing much better than one of the 5670's. It's £20 or so over you budget, though at least you wouldn't need to bother with installing a cooler. From what I've read a card like this will be fine fine for gaming at your resolution, providing you don't mind using medium settings, low AA (if at all). Of course, this will vary from title to title. However, how quiet they are I really can't say with any accuracy, though I think I have read about people using them in dual purpose HTPC come light to moderate gaming machines before.