I think the Dictionary of National Biography gives 1808 as the date of death. It's generally a sound source of such things, though will inevitably have a few errors.
As for the name Willet, it was quite common where I came from for the sons to be given middle names that were surnames within the family, this perpetuating the names. First sons were given the father's mother's maiden name. Second sons the mother's maiden name. I suspect Willet was a surname married into the family.
Now you have my curiousity about "Cooke" as a phonetic spelling. I thought it was a silent E,