There's a reversed-engineerd schematic for it here:
Alesis - MicroLimiter - freestompboxes.org
EDIT EDIT: First thing, make sure the wall-wart is actually putting out 9V AC or thereabouts.
I'm not seeing the bypass circuit, but if we assume it's a true bypass like my MicroGate, the problem will be somewhere on the audio path, which is largely TL084 op-amps.
There's a couple of electrolytics in the path too, e.g. C3 and C16, but it's unlikely they'd both die simultaneously so I would assume it's one of the chips. U1 and U2 are quad op-amps which are handling both inputs and both outputs respectively so if either one of those broke down entirely you'd lose both channels.
However, everything flows through a CEM3381 dual VCA and those are pretty rare, even in the later PA381 incarnation. I'm seeing the chips go for more than I paid for the MicroGate.
Another simpler possibility is that that PSU isn't working right. It takes the 9VAC input and steps it up to +/- 12V. Make sure both power rails are delivering the goods.
If it were me, I'd check the power rails are OK first, and then take an oscilloscope. Feed a test tone into the limiter, and see where the signals goes in and out of each chip in the signal path, starting from the U1 input op-amps, then the Curtis VCA chip, then the U2 output op-amps. Look for the point where it goes in and doesn't come out again.
EDIT: For that matter, is the unit powering on at all?
Also, is anything showing on the meter when you push a signal through it? If so, that would tell us that at least U1, the input opamp is working OK.