4 speed conversion - what went wrong?

The terminal on the starter relay that had a wire that went to the neutral safety switch needs to be grounded. Or, you can replace the relay with a manual tranny one that won't have one.

But it sounds like you are doing that since you hear the relay. If you temporarely jumper the battery wire and the solinoid wire ont the starter it should turn. If it does you don't your wiring right. If it doesn't I would pull the starter and have it checked. Autozone, advanced or one of the other chains should be able to do that.