David Letterman, whose late night talk show hosting career began 31 years ago, has signed on to continue to host The Late Show through 2015. The 66-year-old issued a statement saying, “I needed a little more time to fully run the show into the ground.” Though he has consistently fallen behind Jay Leno in ratings during recent years, he returned to No. 1 when Conan O’Brien briefly took over the Tonight Show in 2009. This February, he’ll have another shot to go head-to-head with a new young rival on NBC as Jimmy Fallon replaces Leno at 11:35 …