The other day I noticed our porch light was out. Assumed it was the bulb,
but that wasn't it, so I removed the fixture, thinking it was a loose wire
nut. There is a black wire and a white wire in the box. Turns out, the
black wire is severed near the back of the box, with just a piece of
insulation holding it to the wire that would connect to the fixture wire.

Is there any way to splice the wire when there is just a quarter to
half-inch to work with inside the box? Otherwise, it looks like I'll need
an electrician to go into the wall. Any advice is appreciated.