Check the polarity on the power cable(s) you're using and make sure that has the same tip polarity as shown at the input socket on the back of your mini-me (the polarity is indicated near the socket which shows the tip being either positive or negative). The mini-me and the mini-mp's have polarity that is opposite from the 'normal' polarity of most other electronics devices, so you need to use a dedicated power cable for your MME that is not used on other devices...preferably the one that was originally supplied with the MME or MMP. On early model MMPs and MMEs, if you plugged in an incorrect cable, you'd fry a diode (and possibly other components) because they weren't reverse polarity protected. It was really easy to do, because the tip size on the MMP and MME are common tips, so what people would do is find a wall-wart from their collection of wall-warts with the right voltage and proper size tip, and proceed to destroy their unit. (That's what I did to my MMP.) On later manufactured MMPs and MMEs, Apogee fixed this issue by redesigning the board to include reverse polarity protection, so if you connect an incorrect polarity cable, the unit won't fry, it just wouldn't power up. I'm not sure, but I don't think there's a way to tell if an MME or MMP is a later reverse polarity protected unit, other than to open the unit up and check the circuit on the board, assuming you know what you're looking for (which I don't). Seems like there is some kind of bridge/shunt (not sure of the electronics term) at the power connection point on the board.
Since you're saying that your unit kinda shows some life and then doesn't power up, your units are probably OK and the only thing you need to do is either purchase a reverse polarity cable, or splice your existing supply and reverse the leads. Before you try this, PLEASE make absolutely sure the polarity of your MME matches the cable...I don't want to be responsible for effing up your gear.
If this ends up working and your units aren't broken at all, I'd gladly take one of your three MMEs off your hands.