You probably have access to another AC cord, swap it your monitor one and retry again. I rarely have any incidents of bad AC cords, far more bad PSUs. Since, you mentioned a 10yr. old system, well can be time to replace if you can find another similar PSU if not a better one should you want to repair. As for accessing any data/files, remove the HD and mount in ext. USB case and/or similar adapter in order to retrieve.
Google for ADA-2020 adapter or locate ext. USB cases that match your HD requirements.
I have an almost 10 year old PC that I've kept limping along, but finally seems to have bit the dust. It simply won't turn on. A power supply failure is the most obvious reason; however, I never heard any noises coming from it nor did I get a burning smell. Instead, over the past week on other occasions when it wouldn't power on, I simply unplugged the cord and plugged it back into a different outlet. I did this successfully probably three different times. So I'm wondering whether this sounds more like a power cord issue rather than the power supply. Yes, I could simply purchase a new cord and try it out, but thought I'd ask first since I've never heard of a power cord not working. By the way, I don't expect to keep using the PC, but it would be nice to access a few things that had not been backed up.