Cables got crossed during a jump and now there are problems! Help? :)

Hello! I have an 85 honda prelude, 2.0 SI. The target for the brake light switch broke,
and the lights stayed on, killing the battery. I had it jumped and took it home.  It sat a
while before I glued in a new makeshift piece that seemed to work. However, I needed
to move my car as per landlord's request. The guy who I asked to help jump it told me
he knew what he was doing, so I let him hook his battery while I hooked mine. Despite
the color-coding, he crossed the wires and melted the cables. Electrical tape fixed
those, but now I'm left with a car that starts (after I jump or charge the battery
overnight), but after I shut it off again and turn it back on, it whirrs and does nothing :( I
got the battery checked and it passed. Any ideas? Thank you, car geniuses.