> What if the person in the room takes a coffee break or goes on vacation?
Regular meat-and-bone people lose consciousness all the time, and regain it later. No big deal.
> What connects the next computational step they do to the previous one and the next...
Whatever index card system or similar the operating procedure in the room prescribes for keeping track of state?
> which then somehow gives rise to a fragment of qualia. That seems like an impossibly complex and unlikely scientific theory.
We don't have any 'scientific theory' of qualia. We don't even know if they exist, or how they would manifest in the physical world.
Since we don't know much of anything, I don't know whether a fragment of a figment would be more or less weird than the figment itself. Or whether we would even have fragments.
It's probably too early to try to have a theory of qualia that would apply here?