What are the chances of autopilot failing to do a flare in an otherwise stable autopilot controlled approach and landing? Has it happened before and how was it dealt with?
In the B767 and B757 the autopilot (autoland) system (triple redundancy) monitors required system components throughout the approach. The flare mode is displayed at 1500 ft and at 45 ft it activates. At any point should the autoland system components (continuously being monitored by the autopilot systems) fail an autoland status annunciator (ASA) and the display directly in view of the crew will alert them of the failure. Based on the weather/visual references they will take the appropriate action (manually land, missed approach, etc.).
I think it is highly unlikely that failure of the flare mode would go unnoticed considering the triple redundancy autopilot system and continuously monitored individual system components (including flight crew active monitoring in accordance with procedural SOP's).
It's a bit more complex than my general description above, but it is pretty straightforward.
In Airbuses, the autopilot always performs an autoland after an ILS approach, unless the autopilot is disconnected by the pilots before landing. At 400 ft the autopilot modes freeze (indicated by "LAND" on FMA) and autoland can only be prevented by disconnecting autopilot or performing a missed approach. The "FLARE" mode activates by 30 ft. Should the "FLARE" mode not activate for any reason, pilots would perform a go-around.
My technical knowledge is not deep enough to tell how it would be possible that "FLARE" mode would not activate after "LAND", except if there would be some critical system failures, like dual autopilot fail or failure of the radio altimeter. In that case, it would also be indicated by a bright red AUTOLAND failure light.
Autopilot is not an autonomous flight control system. In any case in which the autopilot is flying the airplane, there is officially still at least one human pilot monitoring the flight. Unfortunately, that doesn't guarantee anything.
Without considering exactly how improbable it is to ever happen, should such an event ever occur, the likely result is an extremely hard landing, or even a crash and a very angry letter going to the factory.
I don't recall ever hearing of specifically this happening, but for instance the crash of Turkish Airlines Flight 1951 on the 25th of February 2009 in the Netherlands, involved malfunctioning of automated flight control systems during approach. In that case, the autopilot acted too early rather than too late, as a response to a defective ground proximity sensor. That case also proved, that having a full crew awake and present, doesn't stop crucial indications of going unnoticed or noticed too late.