You bought an adapter for it to do what? Is the DD Shager charger actually the "adapter"? Maybe its output pulses with an average voltage of 5V but its pulses have a higher voltage, then the battery is charging at 9V.
The very cheap DD Shager charger sold on ebay has no detailed spec's like its maximum charged voltage and its charging current. It does not say if it detects a full charge then shuts off. Its cover says it charges old fashioned 9V Ni-Cad batteries.
Is your 9V rechargeable battery old fashioned Ni-Cad or modern Ni-MH? Is it half charged at 7.2V (6 cells) or at 8.4V (7 cells)? What is its mAh rating?