If I were doing this, I'd consider the situation a bit; it's likely that there will be a lowish DC voltage that feeds the LEDs but this may indeed be a lot higher than 9V dc because there may be several 3V LEDs all wired in series. I'd also consider that the AC rectification and conversion to the appropriate lowish DC voltage will be better than 80% efficient.
On that basis I'd be tempted to buy/make/build a low power inverter that takes the 9 volts from the battery and converts it to 230 VAC for the lamp.
I'd be interested in knowing the power of the lamp because this power will need to be supplied by the battery. For instance, if the LED lamp were 5 watts, the 9 volt battery would need to supply over 500 mA to the home-spun inverter plus, maybe another 200 mA due to inefficiencies in the inverter.
Ultimately, only you know why you are taking this approach and if I really thought about it properly, I'd buy some LEDs and start from scratch and forget about dismantling an AC LED lamp.
EDIT - regarding the 230VAC inverter - given that it's only going to be used to feed the lamp and that the lamp's internal buck regulation might easily work with an AC voltage that is several hundred hertz, the inverter design becomes a little more straightforward magnetically in that a smaller transformer could be used operating at say 500 Hz. It's also quite likely that the lamp may work efficiently at 200 VAC or maybe a bit lower.
