I believe that the phrase “might want no fact of distinguished die” is confusing because we automatically interpret "die" as a verb. I think the passage is correctly interpreted as involving a noun, perhaps the singular of "dice".
NOUN (plural same)
A small cube with each side having a different number of spots on it, ranging from one to six, thrown and used in gambling and other games involving chance. See also die.
The Jefferson passage in larger context reads:
Determined to keep open a market where Men should be bought & sold, he [King of England] has prostituted his negative for suppressing every legislative attempt to prohibit or restrain this execrable commerce. And that this assemblage of horrors might want no fact of distinguished die, he is now exciting those very people to rise in arms among us
Under the dice interpretation, the confusing sentence
"And that this assemblage of horrors might want no fact of distinguished die, he is now exciting those very people to rise in arms among us"
means something like
"And in order to make this assemblage of horrors complete (like a six-sided die), the king is now exciting people to rebel."
This interpretation is especially easy to access if we assume that "fact" is a misspelling of "facet" or "face". But even with "fact", the interpretation is still accessible given that a die might be thought of metaphorically as displaying six "facts".
The dice interpretation is mentioned at this site.
Unfortunately, it is unclear why Jefferson uses "distinguished" to modify "die," if "die" really does mean "dice." Perhaps the presence of "distinguished" points to an alternate interpretation (see below).
"Die" might also refer to "dye" (as in coloring) or "die" (as in a type of stamping/cutting implement). Both seem more amenable to the modifier "distinguished".
The presence of "distinguished" seems to strongly suggest the stamping/cutting interpretation given by @agc, especially in light of unambiguous phrases like "distinct die" observed by @StoneyB in his comment to that answer.
On the cutting/stamping interpretation, "wanting of the fact of distinguished die" means "lacking officialness", that is, "lacking the stamp or mark that indicates that something (for example, currency) is official". When the king excited people to rebel, the assemblage of horrors no longer "lacked officialness," that is, the horrors became officially endorsed by the king.
Regarding the king who "prostituted his negative"; I assume this means that the king over-exercised his veto power.