You could measure the voltage and current to it to see what the charging circuit is doing (with a high ohm resistor load). Depends on the device, but a common NiCd stack of that capacity, would be trickle charged off 5V rail with a current limiting resistor in series, or whatever > ~4.4V rail is available on the PCB.
If you make up a number, say 8+mA target charge rate on a 5V rail, might have around 100 ohm resistor in series already, so could add another 400 ohms or so across the terminals and measure the current and voltage.
Is there room for a larger battery pack? Since NiMH is less tolerant of the trickle charging, you could offset this by using a larger cell size, like 3X AAA NiMH in a battery holder wired to the PCB, or there may still be some stock out there of AAA NiCd as it wasn't that long ago I saw some cheap generic landscape lights that used NiCd AAA... though I don't know about the quality, if the NiCd are now only made by 2nd/3rd tier manufacturers instead of major brand like NiMH are.
