There is no great way to troubleshoot this sadly that i have come up with. My first thought is if you have another computer to check your router, if it is still functioning its hard to think that it wouldnt fry up that thing yet it would fry up the computer. Not saying it isnt possible because i have seen things that are a bit more odd.
That being said if you had possible lightning hit your house it could have done damage all over because as we all know electricity likes to jump and jump far. The Mac mini that you have has an external power supply this is a plus in this situation because it may be easy enough to take it in to a local Mac dealer and see if that cant just plug it in for you and see if it powers on. If this does not happen to work yes you may have harmed it beyond what i would recommend repairing.
If it is not the power supply it is bound to be the logic board. The logic board of a computer really only has Five main functions and one of those is distribution of electrical power. IF you did get a surge through it one way or another yes it could have hurt your logic board and at this point with the age of your machine i would pull the hard drive get your data and cut your losses. Hope this helps if you get any more info please update me and i will be happy to help!!! Best of luck!!