"Je suis seule dans la cuisine, un chat sur les genoux."
Technically, either les or mes is correct in this sentence.
Mes is seldom used in regular conversation when referring to body parts. To prove the point that both are legal, there is another Duo exercise where mes is used in almost the exact same situation.
The above is based on Sitesurf, about.com and other comments
My guess is that using possessive mes with body parts is seen as getting a little too personal in conversation.
http://french.about.com/od/vocabulary/a/parts-of-the-body.htm "the possessive adjective is almost never used with body parts in French."
Where I grew up, everyone would say "in my lap" rather than "on my lap", especially for small cuddlies like kittens and children. "On my lap" had a sense of unwelcomeness, as in "I spilled a cup of coffee on my lap". Perhaps this should be considered as an alternate? What do you think?