To all who gave advice - THANKS SO MUCH!
I know that the old adapters on the laptop don't seem to recognize the 5GHz signal from my new router, and I know that the new router is capable of both 2.4GHz and 5GHz. However, when I tried to turn on the 2.4GHz signal, I still couldn't get the laptop to link to it. I think with a live Centurylink technician on the line with me, I might have succeeded, but trying to get instructions by chat where the technician can't see your screen was just too unwieldy for me. My daughter was bugging me to fix the machine or buy her another, so I took the coward's approach and bought an ASUS USB Wi-Fi adapter at Best Buy and it worked immediately.
I hope no one is disappointed with the solution!