Oh, that's a pain, I'm sorry to hear that.
I use an
Udderly Ez to milk with, which I think would cope with that situation as her teat would be in the tube. It's a little hand pump (a bit like a breast pump).
Not as cheap as a bucket, but a lot cheaper than a milking machine.
I've used mine for three years now, absolutely no problems with any of my goats, either not liking being milked with it nor with their udders - never had sore udders or mastitis.