If you've had that since childhood then a battery is definitely called for; there's very little chance it's still functional after all these years.
You'll want to verify you've got power going to the battery and motherboard whether you buy a new battery or not. You need a voltmeter set to DC voltage with a range of at least 5 volts. Plug in the AC power adapter and check the output pins; you should see about 4.6 volts coming out of it. If not, replacements can be had for under $5 USD.
Next, I'd verify the power connector and make sure it's passing that 4.6V to the motherboard. Follow the power board replacement guide through step 7.
Nintendo DSi Power Board Replacement - iFixit Repair Guide
Plug in the AC adapter and measure the voltage between the pins on the connector there; the red wire will be positive and the black ground. If you have a good reading on the AC adapter but none at the pins, replace the power board.
If everything checks out up to this point, you've installed a new battery, and it still doesn't work, get back to us and we'll see what your next steps should be.