I have a Lime 2 and
1) 12 volt were applied for 15-20 seconds
2) the POWER button is broken .
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?
LIME2 works with 5V DC only. Applying 12V DC instantly damages it.
In most cases only D3 (overvoltage protection diode) will get damaged. If you are unlucky there might be bigger damage tho.
My advice is to unsolder D3, prepare a card with Linux and put in in the board, apply 5V DC and check if it works. If it works, ensure that 5V DC is used with the board since now there is no protection diode and if you put more than 5V bigger damage would occur. Maybe replace D3 with brand new SMBJ6.0A.
Best regards,
Lub/OLIMEX