I think internal conflict is the way to go.
A member of the council dies, the other four members differ in their fundamental ideology. The council splits either 2-2 or just the majority against any motions. The main issue they are divided over is the replacement of the fifth member, but since the council has stagnated they can't agree to replace him/her or if they should be replaced at all. This would probably devolve into civil war, the members of the council may be old rivals or have unsettled disputes from when they were mere mortals.
Unknown to the rest of the council, the fifth member made provisions for his death, perhaps a candidate to inherit his position. But due to the deadlock of the council and the civil war, these plans were never realized. This would create a role for the PCs, they become the agents of the fifth's will, they stop the war and set everything right again.
Or if one of the their number died, they would realize how little they really knew about existence (a la Socrates) and each would become embroiled in the search for the cause of death and it's prevention. They forget about governance and let the city rot. The task of the PCs would be to recall the council to itself and perhaps make them realize that death may be inevitable.