You can try to burn bootloader anyway, Nano should have the 6 pins ICSP header for this reason. You can't make it "worse",
It's also possible your bootloader is fine, and the issue is somewhere else. If you have a meter, check that the reset pin is being held at 5v, 5v are present on Vcc pins,
and if you have oscilloscope somewhere, check that xtal is clocking. One of my Uno clone arrived DOA because it had a bad crystal.
....which COULD (maybe) lead to corrupting the bootloader. "Throughout history" there have been reports of the bootloader somehow being corrupted by a user sketch. I don't think we've ever tracked down exactly what happens (this SHOULD NOT be possible. But it's happened to me, too.)
Certainly re-programming the bootloader is on a "broken" arduino is worth a try.
"avrdude: Yikes! Invalid device signature.
If you've truly damaged the chip, then the ways it could be damaged are ... very many.You're probably talking upward of 100,000 transistors for the entire chip.
Turning on "verbose" mode for upload (in preferences) will give more details.Usually a signature of 0 or -1 means misconnections or no clocking (but it's hard to tell which, and it could be other things as well. A three-digit value of non-0/-1 would mean corruption of ROM/Flash areas that aren't supposed to be user-writeable.)You CAN do analysis to get a better idea of what happened, but ... usually a board has to cost a lot more than $50 before this becomes economically worthwhile.