I had this exact same problem on a newer laptop. What operating system is the old PC running?
Edit: So you logged off right around the time I posted so you most likely didn't see my question. Basically though, what is happening is the settings for the WAP (Wireless Access Point) are already stored in a wireless profile on your OS. When you changed the settings, it doesn't force each device to 'discover' the new settings of the WAP. So you have to go into the Network connection properties and delete the profile for the WAP. You can then 'rediscover' the network settings of the WAP by re-connecting to the network. It should be noted that when I had this problem, from doing the same thing (changing settings on my WAP) simply deleting and reconnecting wouldn't work as the settings were stored in either the registry or RAM (not sure which) as the WAP has the same SSID... thus recreating the same profile, I assume that it's just importing the settings from the previous profile even though you deleted it. I had to shut down my laptop and remove the battery to clear the RAM (don't know if simply shutting down will do the trick but unplugging the system for about 5-10 minutes will). If you're not familiar with getting into the Wireless profiles to delete a profile, you can follow the directions here
based on the OS. There is another way to 'update' the connection profile settings by selecting the Wireless Profile properties, but the easiest way is to simply Delete the profile > Shut down > Restart > Connect to the Network and rediscover the settings. Let me know if this doesn't work but I'm 99% sure it will.
Edit2: You also mentioned that you enabled DHCP on the router, so you may have the Old PC set to a static IP address that's now inside the defined DHCP range. If the above doesn't work, let me know and I can walk you through fixing that scenario.
Humans are the only creatures that won't live up to the their potential. Give everything your all. When you die, die on E.
So I ask you, what do you intend to do now?