The great standby for questions like this is Transport for London's website.
www.tfl.gov.uk
Use the 'advanced search' option and enter the date and time you wish to travel. Then, in this instance , in the box for where you are travelling from, above 'station', enter "King's Cross" which will bring up a list of options (underground etc). Choose the rail station. In the 'to' box scroll down to 'place of interest' and enter " American Embassy Mayfair". Enter those and the site will list a number of options. It also provides a little plan of the area you are going to, which you can click on to.