I'm a bit too cowardly to use some of the more "windowsy" methods (registry stuff, etc.) but I had a thought: if I get some DVD burning software and install it, will that solve it?
No. If Windows Media Player doesn't burn, other software can't do also.
Frankly said, i belive, something is wrong with your burner itself.
An other
burner SATA 5.25" isn't too expensive, or use one temporarely from an other computer to verify it through the exclusion procedure. There are two plugged cables at the burners back which you can easily attach to the temporarely replacement device that you just put next to the open case.
That would be your next step.
Edit: An other idea is, to delete the dvd-driver completely from the device manager. After that, there should be an unknown device, that will ask you to reinstall. Don't take it from your computer again, take the new one online. This procedure asks you for that.
Good luck.