An L2 fault code stands for Lock Out Type 2, which is a fault igniting.
Attached is the engineer fault finding chart for your boiler, this is certainly something which is repairable, however most likely to be a hardware fault, and therefore need a gas safe engineer to attend and strip down the boiler, I dont believe this is anything to do with the thermostat, most likely a blocked condensate drain or failed heat\flame sensor (which detects the presence of the pilot flame when the boiler starts).
From a consumer perspective the only thing I can suggest is check the condensate drain isnt blocked, which should be a plastic pipe which goes into the bottom of the boiler.
If you would like me to recommend a boiler engineer, please provide your postcode.
s are lockout codes, this specific