That's your starter solenoid, It's a "heavy duty" electromagnet switch, it uses a 12v low current signal from the ignition switch, to make contact between the big positive leads to the starter.
One of the smaller wires is a ground wire and the other is positive, one of them runs through the safety switches of the mower to prevent starting when the blades are engaged or seat is not occupied.
Test for 12v on the smaller wires, one of them should have 12v with the key in the start position, and do it with the parking brake engaged, to prevent a safety switch from messing up the test. Or have someone sit in the seat and turn the key to start while test for voltage. i'd recommend removing the spark plug wire from the spark plug while doing this to prevent any accidents.
I'm pretty sure that the solenoid is your culprit as my cub had that same problem quite a few years ago, replaced it and it's been fine ever since.
i'm not seeing a 115A tractor, but i do see a LA115
A john deere AUC15932 solenoid should get you up and going, This Stens solenoid will do good as well,
https://www.amazon.com/Stens-435-151-Starter-Solenoid/dp/B005LA9UX2/ref=sr_1_3?crid=3J8RNFI22PRVK&keywords=AUC15932&qid=1648568084&s=industrial&sprefix=auc15932,industrial,110&sr=1-3
If you want to someone with more knowledge and experience than me should be along shortly and may give you a better answer.