Easiest way to weld the drill bit to the steel rod is friction welding.
Hold the drill bit in the lathe chuck with about 1/2" of the butt end sticking out, hold your steel rod in the tailstock chuck or if its really long hold in a toolholder keeping the overhang short.
The rod must be lined up with the drill.
Spin the lathe at 2000 RPM and crank the 2 pieces together with some decent force. The instant the joint between the 2 turns into a red molten ball stomp on the brake as hard as you can.
It is often a good idea to take a skim cut off of your new shank but more often-than-not for deepening a hole the runout is acceptable.
Ps. it will most likely be necessary to grind the pressure ridge down that is formed during the welding process.
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