Grab one of Luftek's mini-b power adapters! They're on Kitsch-Bent for $5 (+p&h). They're invaluable!
That'll tell you whether it's a connector issue or not.

Just clean your connectors, if they're gunked up with battery goof juice, no amount of aluminum foil is gonna conduct through that.

Did you reconnect the ribbon cable before putting it back together?

EDIT: Nevermind, I just read the headphone port problem, and that has nothing to do with the ribbon cable.

The aluminium foil didn't seem to work.

I think either the back board is working, an the front board is completely dead (no sound, no LCD, but working CPU), or vice versa.
I'll buy another Gameboy and test each half of the broken DMG's motherboard.

Thanks for your help!

The problem definitely looks like it's the battery connectors. What you can do as a temporary fix (to see if the battery connectors ARE the problem:
Grab some aluminium foil and put it over the battery connectors. Aluminium foil is conductive and will work in the same way as the battery connectors do. A second option is to buy a set of connectors online. See Kitsch-Bent. They sell them for $3.

Let us know how it goes!

So first thing - the line when you turn it off is normal.
Secondly, I don't know what you might have done when you opened up the DMG, but if you say the contacts are greenish, that almost always means leaked battery acid. Clean it off my getting some white vinegar, putting it on a cuetip or other small cotton tip, and rubbing it into the green. It'll melt like wax vs flame. Good luck!

I'm a newbie to Gameboy modding, and recently got an original dmg off eBay.
When I first used it, it worked fine (when I turned it off, a black line of pixels would appear on the LCD, but disappear quickly).

I opened it up to look at its guts, and closed it up. After that, it doesn't power on anymore, and I can't hear any hissing from the headphone ports (with headphones plugged in) or popping from the speaker, as I usually do with other Gameboys.
I did notice the battery contacts were a bit greenish, but they seemed to work before I took it apart.

Any insight as to why it's not turning on anymore?