Are you using a drill press? You won't be able to keep a hand-held drill steady enough to do it with 2mm bits.
By the amount of broken bit sticking out of your block, I can see that have your bits sticking too far out of the drill chuck. You'll get straighter, rounder holes the shorter the bit is. At the bottom of the hole, the tip of the chuck should be within a couple millimeters of the top of your block. Shove that thing farther in there!
Copper is terrible to machine. When it gets hot it gets soft and gummy. Instead of chipping off cleanly, it will stick to the bit and stick to inside of the hole. The result is more friction, more heat, more torque to your bit. Keep your copper cool, keep your bits lubricated.
High speed steel will do just fine for you. Cobalt steel will stay sharper longer, but you'll need to be vigilant with the lubrication because it is more brittle. Carbide is unnecessary unless you plan to make scores of blocks. Titanium-nitride coated bits are nice, because the coppper won't stick to them as much.
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