I am guessing that this is your ENGINE model number:
126K02-
0512-
B1
Using this reference:
http://www.partstree.com/parts/?lc=briggs_and_stratton&mn=126K02-0512-B1&dn=35820003
Check to make sure that the choke item number 97 is not sticking.
Then remove the float bowl nut and float bowl assy. Clean the fuel passages in the base of the carb with carb cleaner including the area where the bowl nut screws into. There is a fixed jet orifice that must be cleaned in the threaded area. Also clean the visible brass tubes in the top of the carb.
Check fuel flow with the bowl removed. You should have a good fuel flow with the float angled down and as you lift it level fuel flow should STOP. If you do not have fuel flow... remove the float pin, drop the float and needle and celan out the hole where the float needle was. Remove the main fuel line and check for fuel flow. Use carb cleaner in the hole where the fuel line connects. If you do not have feul flow from the fuel tank.. check the tank port and look for any kind of in line filter. If you find a filter... check the line before and after the filter for fuel flow. Once you have established fuel flow Reassemble carb
Check the spark.... if you have no spark
First check the thermostat by just disconnecting it. Item 291
http://www.partstree.com/parts/?lc=briggs_and_stratton&mn=126K02-0512-B1&dn=35820004
Then retest your spark. If you have spark with the thermostat disconnected the themostat is bad. (Most engines use a low oil level sensor..... this one uses the thermostat)
There is an electrical contact on the throttle plate that is grounded when the throttle is selected to off. Make sure the throttle plate lever moves off of the contact when you advance the throttle. ALSO... there is another ground / contact on the engine brake. Make sure that when the engine brake is released that the brake lever moves away from the ignition grounding contact.
The above are the basics.... If after doing ALL of the above you still have problems with
ignition then suspect the ignition components.
Thanks for choosing FixYa,
Kelly