How exactly does the oxidation of TEMPO by NaOCl/KBr work?
I would propose the following mechanism:

Note that the oxygen radical anion isn't that bad as a leaving group as the Cl-O bond is weak. It can participate in an electron transfer with chloride anion, or rather the bromide at the beginning of the reaction as it is more abundant (I have written the protonated form - hydroxyl radical as it is less likely that two anions will come close enough for electron transfer):


The bromine radical can in turn react with another TEMPO molecule. That is why two equivalents of TEMPO react with one equivalent of hypochlorite.

Keep in mind that there are countless possibilities and deviations from the mechanism I proposed.