Turns out, I have an overactive thyroid. And possibly Graves' disease (they're still testing for it).
So now I'm on medication: something to bring my heart rate down (and it does seem to work!), and something to stop my thyroid from producing any more thyroid hormones. Then in three weeks, I'll get other meds to mimic those thyroid hormones; and I have been told I should be taking those for the next year to year and a half. Then, we'll see if I can get off them.
The takeaway from this for me is: you don't lose weight without putting in any effort :-(.