Mk160 power suply

can I use a battery instead of a power supply: 12VDC / 100mA ?
If yes… what voltage?

You can use a 12V battery. Max power consumption is approx 60mA, this allows you to calculate battery capacity.

I built a kit that worked fine till I tinkered too much with it; the LED doesn’t light up, the relay doesn’t click, even if I short the LDR–can’t find the source of the problem now. Possibly I switched the polarity of the 12V power supply and kept that for some time (I was busy on the phone in the meantime). Could wrong polarity have destroyed the kit?

Reversing the polarity more then likely damaged it

You should start by checking for voltage between pins 1 + and pin 8 - of the IC

Please check the diagram: Diode D1 protects the unit against polarity reversal.
There must be another reason for the malfunction.
Sorry, this forum does not allow picture uploads
If you can provide LINKS (e.g. by using a free online picture base) to HI-RES and SHARP pics of your assembly (both solder- and component side)
then we can take a look and provide some feedback.