How are fuel injectors in a jet engine monitored to detect malfunction?
Fuel nozzles are overhauled and bench tested to meet the manufacture's operational limits.  They are controlled for both flow (back pressure) and also flow pattern.  The FADAC monitors fuel flow, this along with the compressor vane position are the main input controls for thrust generation.

The fuel nozzles do degrade during engine's operation.  This contributes to the engine running hotter as it ages.  Other contributors are the erosion of the compressor hardware and the burning/oxidation of the combustion chamber and turbine hardware.  In fact a defective fuel nozzles may lead to a "hot spot" that accelerates the burning/oxidation of the combustion chamber.  The FADAC and flight crew will monitor the throttle response and also the Exhaust Gas Temperature (EGT) as the controls to understand the relative health of the fuel nozzles.  If the engine exceeds the EGT limit a mandatory boroscope inspection is required to keep the Aircraft in service, this inspection will visually verify the condition of the fuel nozzles and combustion chamber.