The only thing I can think to do now to recover the job is to either buy
a 13 x 250mm taper drill and use that to get a good clean pilot hole to
guide the core in, or grind down an sds shanked 13mm drill I have here
to fit the arbor and use that as above. 3rd alternative would be a bit
of 10mm silver steel rod and grind a taper on that to fit the arbor to
get the core started clean.

Try a wooden plug in the pilot hole. You'll be surprised how an sds bit
is able to push into wood nicely.