I have the same issue. When boombox is off, it draws around 22mA, when I turn it on it's around 42mA, seems like normal standby current, but does not seem to charge, although it turns on and draws battery power as well as it works when plugged in as current draw seem to fluctuate with sound level and goes well into 2amp territory when I crank up the volume.
I did not manage to find boombox service manual, although xtreme service manual is available. As xtreme does not have any kind of battery gauge, so it's very likely that boombox uses similar system. Xtreme uses TI BQ2461X battery charge controller which should start charging if battery voltage is above set threshold and it is not overheated.
I disassembled my boombox trying to isolate the problem, well I didn't have to look far, picture attached. It's BQ24617 standalone battery charger that's fried, will try to replace.
Edit: I've replaced BQ24617 and R97. Seems that the unit was connected with reverse polarity. Charge controller has no reverse polarity protection designed for it. Works correctly now, chrage current around 1A+ and reducing approaching to full charge.