transfering from Heathrow airport to London City airport

Here's the way I'd do it: sounds a bit complicated but has been picked for fewest stairs/escalators: Piccadilly Line tube from Heathrow (escalator down from terminal) to Hammersmith. Walk across to the other side of the platform to get a District Line train in the same direction. Get off the District Line at Westminster and go down the escalators to the Jubilee Line eastbound. Take the Jubilee to Canning Town and take the lift up to the Dockland Light Railway. Take a DLR train that says Woolwich Arsenal on the front to City Airport station. Looking at the map you might think it would be easier to change from the Piccadilly to Jubilee at Green Park, but DON'T DO THAT. It's a notoriously long walk. There's no way to reserve tickets in advance, but you can get one easily at the Heathrow Airport underground station.
Or if you can afford it you can get a taxi.

Thanks! I have NEVER been in England... I could try this, if I felt adventurous enough... but I don't! I was wondering if there was a shuttle or bus.... looks like I will pay for a taxi!
Thanks again.

If you don't want to take the Tube and the DLR, book a private car service like Just Airports. It will be a lot cheaper than a taxi. Your biggest potential problem will be the connection time. If you arrive late at Heathrow and/or are delayed getting through immigration/passport control, you stand a very good chance of missing your flight at London City. I'd want a minimum of five hours to make that connection.

Additional note: To save yourself the connection hassle, you could fly JFK-Paris CDG and CDG-Florence on Air France.

Taxi fare from Heathrow to London City Airport will be between 90 & 100 GBP ($145-165) and take anywhere from 1:15 to daysdepending on traffic. Yes, what time you arrive will factor in on your time. To go between these two airport you literally go through all of London and traffic can be murder. I personally do not think 4 hours is enough time to land, get through immigration & customs, grab a taxi, drive to London City, check in, go through security, etc. Considering what the taxi will cost....you might want to look into flights out of Gatwick which is easier to get to...and cheaper.

Thanks to each of you for the perspective... reality check... I won't do this again... Thanks to Rick Steves for this forum !!!! I checked about changing my Air France tickets but it would cost almost the same as buying a new round trip ticket... sigh. I have emails into Heathrow Express and the other suggested airport chauffeur service.... I am breathing and praying!!!! I think it will work out.
BL