CMM1200 Type 1 mower won't start

My CMM1200 Type 1 mower won't start. I read a few of the issues posted on this site that were similar but I'm not certain if this is strictly a dead battery issue, a charger issue, or possibly both.

I kept the mower connected to the charger when not in use (I'm in California so we do use it year round).
The green LED on the charger was always lit so I expected that the battery was charged.
I just measured 5 volts across the top cell and about 2.5 volts across the bottom cell with an inexpensive analog volt meter.
If the green LED on the charger is lit, does this mean that the charger is no longer functioning properly due to the low voltage across the battery cells? , ...., and should it be replaced along with what I assume to be a dead battery. I'm pretty sure that the battery is over 5 years old, as this mower is a hand me down from a friend when the switch lever on my old B&D MM850 broke.

I'd hate to purchase and install a new battery if my existing charger were part of the problem as well. Any thoughts on how I might take this a bit further to see if I need to replace the charger as well or other things I should consider here?

Thanks, Tony

CMM1200 Type 1 mower won't start