I would strongly suggest an inline fuse on the positive side of the power supply. I'm doing this from memory so anyone feel free to jump in and correct me, but I think I'm pretty close on my estimates. It's not surprising the 22-24 gage wire melted first. It would only be able to take maybe 5 amps max of load depending on the number of cores before heating up, and a jammed wiper motor would be attempting to pull way more than that.
As a general rule single core wire can handle more load. For example; 12-14 gage wire stranded can typically handle 10-15 amp, and single core 25-35 amp load. So a lot depends on the quality/type of wire used. I would look at the output rating of your power supply and adjust the fuse rating to that so you don't accidentally cook your PS.