A family of princes was then on the British throne, whose treasonable crimes against their people brought on them afterwards the exertion of those sacred and sovereign rights of punishment reserved in the hands of the people for cases of extreme necessity, and judged by the constitution unsafe to be delegated to any other judicature.

It's obviously the Stuart kings, James I of England and his son Charles I of England - the latter of whom was deposed and murdered by his own Parliament. Charles is even mentioned by name a couple of paragraphs later.

King James VI of Scotland became King James I of England in 1603. A few years later, the first English colonists settled in North America. They even named the river they settled on "James River", after the king; this settlement became known as Jamestown.

James' son Charles was deposed in 1648 and executed the following year.

The timing of settlement matches the Stuart dynasty. The "exertion of those sacred and sovereign rights of punishment" referred to in Jefferson's passage clearly refers to Charles' deposition and execution.