Best way to make wiring to outbuilding legal
Oh my gosh! It sounds like you have a 120volt sub-panel if both the hot leads are jumpered.  That's an OK practice for ppl that don't need a lot of power or 240 in the sub-panel.  

Did you really say the "other black" is really the neutral and "capped off"?   That means the ground is being used for the neutral, very dangerous and very illegal. 

The good news is this isn't hard to fix.  You should:

1)  Turn off power to the sub-panel NOW. 

2)  remove from the neutral and connect the bare, unshielded ground to the ground bar.

3)  Connect the neutral to the neutral buss bar (be sure to code-tape it with white tape), make sure the neutral is indeed connected to your neutral buss bar in your main panel. 

4)  Make sure the neutral and ground are isolated. 

Assumptions:  that your 30-amp breaker is a single pole breaker, or that only one pole is connected. 

Alternative:  If you want 240 in your shop, you'll need to run another wire for the neutral, remove the jumper between the 2 hots in the sub-panel and connect the wires properly, You may also have to swap out the 30 amp breaker for a double pole one, depending upon what you have now. 

Your "handyman" missed a typical issue with sub-panels, you are required to have a neutral back to the main panel.  You won't have that if you go 240v unless you add another wire from the house to the shed.   You would have a separate neutral and ground if you stayed at 120 and made the other changes. 

You are right, this is monkeyed up and congrats for realizing it and fixing it.