Hi Julian,
The Bike Shed is probably worth a visit but The Ace Café is a long way from the City - out near Wembley, although there are two bike meets during your stay - check on their web site.
There's not many bike shops in the centre of London, they're mostly in the suburbs. In-Moto Croydon is one and Ray Petty is possibly the nearest to the centre of London, which is located near the O2 which has shops for the lady.
Re. Markets, in the City there's Leadenhall, Spitalfields and Borough Market. Also Brick Lane on Sunday is worth a visit. All in walking distance from the Bike Shed
The South Bank is also worth a visit this time of year, plenty of market stalls & places to find a beer or two.
Hope the weather is kind to you!
Nev