There is a home produced alternative David, especially as the tap is not too deep.
You could make a hollow cutter from silver steel and providing you can position it accurately over the axis of the broken tap you can mill the metal from around the outside.
The OD of the cutter should be as small as you can get away with perhaps 5mm in this case. Cut the teeth with a triangular file and H&T. It is a slowish process but by it’s nature quite controlled. As you are quite shallow you will probably not have to go far before the tap will loosen. The cutter does have to be continually withdrawn to clear the swarf and of course it does leave a larger hole. Don’t debur this hole. Loctite in a small slug and peen it over then machine off before re-drilling – possibly you still have plenty of depth of the original hole left to take the strain.
I have used this method – it does work and has it’s drawbacks as stated but you need nothing more than a bit of silver steel and the means to harden it.
Good luck which ever way you choose – it can be a very awkward set back but not unsolvable.
Hope this helps in some way,
Regards – Ramon