Beware if you replace the whole thing that the new one fits. Apparently the "replacement" is a bit bigger and you have to do something or other (forget) to get it to match in back with the ducts (or something like that! :( ) Huge PITA.
You said you get a click for the igniter after the fan is on, so the sail switch works. If you get a click but no ignition, maybe not enough gas, but you said you got a brief burn, so there must be gas. So it is all about keeping the flame lit.
The sparkers are supposed to sense that, but you cleaned them. ISTR on other similar issues there is a "ceramic" thing that might screw that up (but that might nave been on a water heater or fridge?)
The board might be the problem, but if it lights and won't stay lit, it could be the electrode sparker thing at its end, and not the board.
Sorry for the lack of technical terms. It is what it is. (Ours is an 8531-II, so older, might be a little different.)
