I cut the broken 4-pin connector off and found there are only two 18 AWG wires.
I'll see if I can find a female connector that will fit the male ends on the Segway and just be careful to hook it up correctly until the 4-pin connector arrives.
EDIT- Just found this image;
I measured the male end of the connector at 11mm diameter. Since I don't see a 10mm version of the plug I'll assume it's actually 12.
I ordered a GX12 connector from Ebay, but it will probably take a month to get here from China. $1.59 shipped to my door if I'm willing to wait.