AC vs. DC power supply question (non hi-fi related)

My son's got this nightlight that went dead, and he loves the thing. I'm trying to get it working again. At first I naturally thought it was just the bulb, so I changed it - no help.

Inside is the light, and a 4-colored circular filter attached to a small motor. The motor turns the different colors spin on top of the light, so the light being emitted is changing colors.

I threw the power supply across my meter, that's the culprit. No power. It's just a transformer that plugs into the wall and has 18ga coming off of it. I looked at the rating on the power supply and bought what I thought should be a compatible power supply.

However, I just noticed that the original power supply says that the output is 12VAC. I've never heard of one of these little things putting out AC, I thought these things are DC?

I just tested the new power supply, and I'm getting strange results - the light comes on, but the motor doesn't spin. Would this be symptomatic of the need for AC power?

I'm just wondering if maybe the cheap Chinese power supply is mislabeled somehow, and maybe it's just that the motor is no good? Otherwise, I'm stumped, I can't even seem to find a 12VAC supply online...

Thanks in advance...