Yeah you should take the bike, 7 weeks is a Long time and hopefully this May will be sunny. You're gonna hate being without a bike. Chelsea is a pretty afluent place but it's also pretty congested with traffic and the roads are bumpy. Kings cross has a large train station and a few other stations in close vicinity, also congested. But it's quite central to londons shopping centre though I recommend you take the subway rather than the bike.

Chelsea is close to Knightsbridge, home of harrods.

Good luck with your trip, I have to apologize though as I won't have time to meet up (wish I could but I'm just being realistic due to family and various other old-age stuff)

Edit2 bloody iPhone and it's spelling mistakes

Edit sorry I might have given the impression that biking isn't fun in London diego cngestion and boy roads. It is a lot of fun however, Hyde park is just down the road from where you'll be and that's a real nice place but you have to ride slow. Like the others mentioned, you could grab a train and get out of London. I've never done that, all m riding is within London and on the outskirts. So bring your bike you'll have a blast!