Does a period, semicolon, comma, or nothing go there?

Solution 1:

A Google search for "He's quite dead. I assure you." (which as expected shows more hits than the "I'm" version, if not many more) only gives examples of your versions B and A.

Comma splices are not wrong per se, as discussed in this previous thread. And I'd say that version B is the best here ('I assure you' is best not analysed as an independent clause anyway).

Version A 'reads' wrongly, to my ears: the natural pause isn't signalled.

Version C treats 'I assure you that', essentially, as an independent clause, and is clumsy at best.

Version D is fine if you want a more dramatic pause.

Version B treats 'I assure you' as a modality marker, a parenthetical, which is probably best set off by the comma.