I've managed to install the game and everything from the new Daggerfall download without a hitch, but whenever I try to install the patch it goes to "Insert Daggerfall CD-ROM", and won't let me go any further. I have looked over this forum and haven't found anybody with teh same problem. Then again, I am notoriously terrible at searching.

Can somebody help me with this problem, or provide a link to somebody else's solution?

You just need to start DOSbox and go through the steps for playing Daggerfall except for the last command that actually starts the game. Then try installing the patch.