In New York there is a kiosk in Time Square that sells same day, discount tickets to Broadway shows. Is there anything comparable in London?

Yes, TKTS in Leicester Square. Don't be misled by the shops offering tickets you'll pass on the way there. You want the real thing.

http://www.tkts.co.uk/leicester-square/

Check the closing times. Don't make it a last minute thing.

Discounted tickets are slso sold at the box offices of sh
ows for the day of performance

You may be surprised to find front-row seats still available well after the tickets office has opened. Then, when you sit in the seat, you definitely will be surprised that you are below the lip of the stage with a fine view of the performers' knees. Some bargain. Middle-of-the-orchestra rows are far better, and far harder to get.