You need a new battery. Kohler engines are reliant on a good battery. Take fine sandpaper or emery cloth and clean off the flywheel magnet. Then regap it. Storing the machine/engine in a damp/humid environment will kill a Kohler engine. I know, because I am a certified tech for them now 12 years going. I do need updated... Use 89 octane gasoline. No E85, or 87. They are/have alcohol and bad corrosive solvents in them.
I suggest you change out your spark plugs, gap them properly. Those coils (magnetos) will often have replacement spark plug wires. Those will short out over time, or become loose. I prefer to use dielectric grease lightly on the connection points. Check all of your wires and connections for corrosion, clean as need (make metal shine). Grounds, hot, and cold. Should crank right up good as new.
My Kohler SV 740 Courage Twin engine recently started missing on one cylinder, finally failing completely and then a few days later, failed on the other cylinder. Troubleshooting, I found it would not produce a spark at either plug . Checked the magneto coils, gaps, etc, with no results. Assuming the coils were defective ( a strange thing for both to do) I replaced the original coils with the replacements for that engine, gapped them at factory settings, STILL NO SPARK ! Would I be wrong in assuming the flywheel magnet is the problem ? It will hold a small screwdriver but seems weak. Anyone had this problem ? Thanks.