I have only been in the area a relatively short time and am looking for somewhere to play badmington. Doesn't necessarily have to be a 'club' just a court that's for hire by the hour would be fine. I am based on Upper Ground near Blackfriars Bridge.
By the way, really enjoy reading your forum, you'd be surprised how informative it is to a 'newby'
Probably the most convenient place to you will be the Colombo centre (Colombo street about 2 minutes from Upper Ground) which has a hall with three courts. It can be difficult to get a booking in weekday evenings but easier at the weekend.
Colombo is quite quiet at weekends and costs 6.90 for an hour. However on top you have to pay a £1 entry fee per person if you're not a member. Membership is really cheap at the moment as a local resident - £90 for 6 months. They only have a gym and courts and the changing facilities are as you'd expect from a local authority joint but it still beats the usual extortionate gym prices and it's really close to you.
Oh and they also do Saturday tea dances - don't know how old you are. . . . .