Unless you use a hard-surfacing rod, the reason the weld is hard to drill is because it got hardened by rapid cooling.
When you arc weld a part, the heat is confined to a small area, and when you quit welding, the rest of the mass sucks the heat out fast and makes the weld hard.
This is also the reason many welds crack just past the weld. There is a very thin line between the really hard metal and the softer metal next to it. This acts as a stress riser and concentrates all the stress on that area.
All you have to do is anneal the weld.
After you are through, grab a rosebud torch, or whatever torch you have with a wide tip.
Heat the weld up until its cherry red, hold it at that heat for a few seconds to let the heat spread, and then let it cool slowly out of any moving air.
It should drill just fine after that.
If you are worried about having drawn the temper of the steel, just re-harden it after you have drilled it.
Polish up a section, heat it slowly with the torch and watch the color of the shiny part.
When it turns a deep blue or purple, let it sit for a few seconds, and then quench it in a bucket of oil.
If you want it harder than that, just let it continue to change colors until the blue/purple starts to lighten up toward a straw color and then quench it.
If you don't want to waste the oil, or if the smoke and flames are a problem, then use salt water.
Its not too thick, so you might get away just fine with fresh water, but why take the chance?
