Well, this is hardly definitive; however, I have not been able to find any quote that was attributed to Einstein concerning him sleeping 10-11 hours a day. Also, I have found no reference to it that gives any sort of citation as to where they arrived at that information. One example of this sort of behavior is the book Power Sleep. While it makes the claim that "Albert Einstein claimed he needed ten hours of sleep to function well," no citation or reference is given as to when Einstein claimed this. This book, however, is listed as a citation on at least one website: http://library.thinkquest.org/25553/english/basics/need/amount.shtml. Various other news outlets use the same information without citation like the linked BBC story in the question.
The only quote that I have found concerning sleep that is attributed to Einstein is on that goes, "We sleep 1/3 of our lives away." This would imply that he accepted the idea of sleeping 8 hours a day (8=24/3), but again, there is no citation.
All in all, what I can say with certainty is that the majority of references to Einstein's sleep are drawn from other secondary sources that themselves do not give any evidence for this being the case. Many of Einstein's letters are available, but I have not had the opportunity to pore through those.