5 what? AMPS?
They usually show the amperage that the battery is drawing to charge, so a lower number is a better charge. I think 5 is too much. After an overnight charge mine would be just above 1.

Crappy tire is good for testing.

It does sounds like the battery is not holding a charge.
You could check this pretty easily with a voltmeter.
Test the voltage with the battery off the charger. Since it was just on the charger it might be a bit above the normal ~12.5v, but wait for it to drop down. If it keeps on dropping down much below 12.5v, then you've confirmed that the battery is toast.