You can more or less forget drilling it out Brian. A carbide drill might do it but would probably snap. You could try drilling down the flutes with something in a Dremel but if it tries to remove material from the tap you will probably finish up with something else stuck in it. The perfect answer is spark erosion.
Looking at the photo I would have thought a decent quality pair of 8 or 9" engineers pliers would get it out. Mole grips too as pgk suggested. Try and find a old major brand name.
Tapping M2 is not like M6. It needs more care, tap square and more frequent reversals to break the chip.
Usual reasons for breaks.
Tap not square to hole – I suggested that you should check to see how square your drilling machine is to the table and adjust if needed – a large 300mm adjustable spanner will undo the nut. They can usually be bought in sets of 3.
Not reversing often enough. People are inclined to never do that often enough and get away with it until the try very small sizes. It's obvious really. Think about the width of the flutes and how much of a turn is needed before the chip has to bend. The tap will bend the chip but it's easy to ask it to bend too much.
Wrong sized hole – check the sizes of drills. Don't assume that they will be correct at small sizes. Brass tends to cut very cleanly so this may well be your problem. Taps at this size shouldn't take much effort. If you measured a 1.7 rather than 1.6 you might find it's 1.65 so that would be the one to use unless the 1.6mm was definitely 1.6 dia or slightly larger. You'll probably find it's undersized. They usually are.
Here is a source of engineering tools for you
**LINK**
Those pliers are ajohn approved but sadly lack the notch in the side for cutting thick wire.
If you want to get some locking pliers the rs pro ones should be ok
**LINK**
Their adjustable spanners are a bit expensive. As a company they tend to sell high end and ok stuff and no junk. They did sell Chinese lathes at one point but dropped them – too many complaints.
I use a set of ones that are similar to these also the plain ones without plastic on the handle. The main thing is the material they are made of
**LINK**
John
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