If you are rich enough to take a cab I'm surprised that your driver didn't just bring the Roller. Skip the cab take the tube, faster and much, much cheaper unless you want the real London experience, particularly in that part of town, of sitting in a cab, going nowhere and watching the meter tick up another £3, then another and you've traveled 200 metres. A cab would probably cost around £20 depending on the time of day.
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Keeping London moving (you might not). You can use a payment card to get through the barriers. If you get stuck just ask somebody Londoners aren't as scary as they look, especially the ones in LT uniforms. I'd find it more puzzling to go to Cheshire and take public transport.