Divorced life just got sweeter for cutie-pie chef Katie Lee. Billy Joel‘s third ex has just sold her West Village townhouse for $11.65 million — nearly double the $5.9 million the Piano Man paid for the Perry Street pad in 2006. The mystery buyer is the New York-based head of a multibillion-dollar finance company who is “fiercely guarding his privacy,” sources tell The Post’s Jennifer Gould Keil. Let’s see for how long. Lee declined to comment when asked what she’ll do with the profits from the sale.