I'd like to drill a 1 5/16" (1.3125 inch)
hole in a 1 inch thick mild steel..........
It doesn't have to be pretty, and the jig I'm
making doesn't require super
high precision.

Here's one that some machinists will recoil at, but with a steady hand a
good result can be gotten:

Cutting torch, fine tip. Turn the material on its side and pierce a hole
through the middle. Then clamp the material flat with the hole overhanging
the edge of a welding table, and carve your hole to its final dimension.
Dress it with a high-speed rotary rasp.

Fast, lots of fireworks, cool hissing sound as the steel burns. Voila!