I have a Daniel Burket that might be a match.The info comes from the pamphlet "Descendants of Jehu Burket," published in 1940.Here's what it says:
"1. Daniel Burket, 1818-1919.George P. Burket, San Jose, Calif., and Mrs. Amos Graybeal, West Jefferson, N. C., are children, and Rev. Charles C. Weaver, of the M. E. Church, Winston-Salem, N. C., is a grandson."
This Daniel is one of two children of David Burket (1797-1831) and Catherine Doolhyer.The eponymous Jehu Burket is Daniel's great-grandfather.
I don't have anything further on Daniel, so I can't confirm his wife's name or the names of any other children.If this turns out to indeed be the David you're looking for, let me know and I'll send you more information on this Burket family