The engine in it should be a Briggs 42E707-2631-E1 if original.

I'd suggest downloading the IPL for that engine from the Briggs website so you can see the exploded views etc..

What I would do is remove the starter motor and clean/lube the spiral Bendix threads on the drive.

IF they are gummy, the drive can't move forward fast enough to engage the flywheel teeth when the starter motor starts to spin.

See if you can turn the stacked pulley on the bottom of the engine, under the tractor. If you can't do it, then try turning the flywheel. you might have to remove the flywheel housing-3-4 small bolts, and try to turn it. If the battery isn't charged enough, it won't spin the starter enough to make the bendix rise up to contact the flywheel. Also, if you got the battery connected backwards, the starter bendix gear won't rise up to contact the flywheel!
Plus cable (+) from plus post on battery, to solenoid. Minus cable (-) cable to frame/ground, or engine block for the "Ground".

Havent turned it on in say a month" Dead Battery??? That would be my first clue to solve. (you would solve or verify the battery good by charging and load testing or take it to autozone for free test). Next would be the connections at the battery, starter, and frame (grounding point). Another tidbit of information would be How old is your Lawn tractor and battery? This may give us some clue to what may or may not be problem.