What is the best (shortest) way? Having alredy flown between London & Eindhoven, I would like to experience the train.
If you want to go to Eindhoven rather than Amsterdam, take the Eurostar from London to Bruxelles-Midi and IC trains from there to Eindhoven. You will need to make connections between Bruxelles and Eindhoven and the route varies depending upon your departure time. To see this, go to www.bahn.de and click on the drop-down flag menu for English. After you get summary timetables for Bruxelles-Midi to Eindhoven, click on the arrow to the left of each departure time to see the connecting points. On that same page that shows connections you can click on "Show map" for each route.
There is no advance purchase discount on IC trains. Buy your tickets when you get to Bruxelles.
Follow Tim's recommendation. No one knows trains better than Tim or Lee.