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It will be much cheaper to travel by train, tube and bus and probably faster too. The website below gives a fare from £82 City to Gatwick (which could be much higher depending on the traffic). It also says that they accept visa, but you should check individual cabs before getting in. The journey would take around an hour and 10 minutes (on a good day), but could take much longer with bad traffic.

To do the journey by public transport, train from Gatwick to London Bridge costs £9.40, tube from Monument station to Tower Hill will cost about £4. That is under £15 for the whole journey and will take around 45 minutes.

For all transport in London, see the transport for London website. It has journey planners and maps and information on fares and tickets.

If you use your card, you will be charged the exchange rate plus any other amount your bank sees fit. Try to use cash where possible, especially for small amounts.

Some black cabs accept cards. They will have a sticker in the window if they do and you usually have to tell the driver in advance that you're using a card.

Getting from Gatwick to Tower Bridge will be much cheaper and faster by train (train to London Bridge and then walk/tube/taxi to the hotel)