I tried applying the scale with CtrlA > Scale but it did nothing. How do I get the edges to bevel evenly?
Work on your Pre Beveled mesh, or undo the bevel.
You have double vertices in three of your corners. (remember I'm talking about the unbeveled mesh.)
Select everything (AA) and Remove doubles found in the tool shelf or from the specials menu W.
Then in edge select mode you can reselect the top edge with AltRMB , and use the bevel tool.