If it is dead cells the battery will not fully charge, just develop a surface charge, Battery needs to be charged and loaded test at one your local Automotive parts stores as they do it for free.
Also once the engine is running you need make sure it is charging the battery at 13+ VDC. If note it is likely the rectifier diode in the stator lead is bad. They can open, short, or become leaky. These stators are repairable if just the diode is bad. otherwise you would need Kohler PN 12 085 08-S which list for $85 but most places are higher especially JD. Besides I can't even pull it up at my JD dealer under either of the two PN they gave me on JD Parts.
Now this assuming you have the three stator as JD Parts does list the 15 amp setup too which incase you have the voltage regulator setup but I went by the engine's spec number posted which is the 3 amp setup.