I have a Lime 2 and
1) 12 volt were applied for 15-20 seconds
2) the POWER button is broken (https://ibb.co/N9WWhfd).
I try to power it up using the battery or the power jack, but the LEDs do not light up.
Should I consider it completely broken/unusable or is there any other way to check?
Probably part of the power circuit got damaged. LIME2 is 5V only board and 12V would damage certain parts.
If the board doesn't get powered by neither PWR_JACK nor battery over the LI-PO connector then probably the AXP209 got burned. Inspect the schematic for what might have got damaged on the way in.