Don't use Merge. Pathfinder operations such as Merge are meant more for shapes, not paths. Pathfinder isn't really aware of strokes and doesn't attempt to combine or alter strokes as well as it should. Essentially, only use Pathfinder when you want to alter full shapes, not merely strokes.
Simply select the two end points and choose
Object > Path > Average set it to "both" and click okay. Then choose
Object > Path > Join. You will have one corner point. You may need to move it back into the proper position after this.
To average and Join in one step, you can use a shortcut. Select both the end anchor points and hit Command+Option+Shift+J (Mac) or Ctrl+Alt+Shift+J (Win). This will average and join the two points in one step. There is no menu item to do this, only the shortcut will do both at once.
There is also a free script for Illustrator to round any corner which makes the entire processes of what you are trying to do much, much easier. And there's the VectorScribe plug in for Illustrator which contains a feature to allow dynamic corner rounding (this is what I use).