I've never played a Muslim faction, so I can't answer from experience. However, I'd guess that's like an uneventful crusade. The same thing probably would have happened if Edessa had been captured by an army that wasn't part of the jihad. You technically didn't capture the city. You gained ownership of the city because you were besieging it and suddenly there wasn't a garrison anymore. If you had assaulted and won, it would have been a successful jihad.

From my experience (on the receiving end ), the jihad ends when the last army that is part of the jihad is destroyed.