It's been a while! However, the research has never ceased. I've recently been retracing some members of the family and came again to William K. White.
I'm sure you're aware at this point (perhaps not, and if not, perhaps someone else can benefit) that William K. was a Civil War Soldier, and enlisted on July 20, 1861 by Colonel Bohannon. He was discharged by reason of age. He married Susan A. (Susanna) Down (b.1831, d.1910). They lived in Mobjack, Virginia on a peninsula known as White's Neck. When I visited for the past 3 years, there are many references to the Whites in this area, including a cemetery called Providence, which has many White members buried within.
On April 8, 1871, William buys 6.625 acres from Joshua Gayle and Sarah E. his wife. The property adjoins his own (William's) and included everything except the fence, which Joshua was supposed to move "when convenient".
William's parents were Dudley White (b.Sept. 30, 1780, d. 1853) and Rosanna Hudgins (b.Jan 6, 1791, d.1860). Dudley's parents were Edward White (b.1751, d.1809) and Pamela (Pembroke) Singleton (b.1762, d.1813, m.1778). Edwards parents were William White (b.1709, d.1796) and Elizabeth Bartlett (b.1715, d.1760., m.1749)
Above them is speculation at this point. There are many William Whites which people (in my opinion) fantasize about tying to William White of the Mayflower.
Another William K. White is also in the area, known as William Kitson White. William Kitson is Dudley's brother, and also a son of Edward and Pamela, and would be your William K's uncle.
Hope this helps!