You could try stuffing lots of oily rags (dry rags are not as good) in there and a use a strong magnet close by to attract the filings while drilling and then keep swiping the magnet around till you get it all, you only need to use a 5 or 6mm drill then use a bigger easi-out, with the heat generated, this should come out easy.
Failing that, it is head off at least
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