Hi Sark,
You have good taxi services from LGW. However as they are accredited to work from the airport they charge you a lot.
There is a more simple way which is to go with National Express. They have coaches going up and down all day.
Have a look at their web site. You can book online and if you miss your coach because you were delayed they will take you anyway.
The journey is 1 hour.....if the traffic is good.