I looked into this topic in depth, and we had a great discussion about it
here last August and Sept. My cat, now almost 11, was diagnosed with
hyperthyroid last August. The radioactive iodine treatment is one of
two cures--and the best and least risky of the two. The other is surgery.
Treating with pills such as Tapazole amounts to controlling the disease
but is not a cure.

Radiotherapy is a simple injection, no general anesthetic, and a
90+ % cure rate. The only down side is you have to leave your cat
at the vet anywhere from 3 days to over 2 weeks depending on the
laws of your state regarding the handling of radioactive waste. (Because
your cat's waste is radioactive for a little while.) It is expensive: about
$1200.

My vet was ready to do the surgery, but Phil P. and Maurice and
others talked me out of it for very good reasons. For one thing it
is just a lot more invasive to the cat than the radiotherapy--major
surgery that involves general anesthetic. For another, it can be
challenging to ablate the thyroid while leaving enough of the
parathyroid. Needless to say this is tiny in cats. The cost is
about $800.

Instead, I chose to treat with the Tapazole until her levels
of thyroid hormones are stable, and to have her tested for
kidney and liver function, then if I got the go ahead to get
her the radiation therapy. Why test for liver and kidney
function? Because the cats must have healthy kidneys
to undergo the radiation, and they must have healthy livers
to take Tapazole. Why get their thyroid levels down with
the Tapazole before you test? Because since everything is
speeded up in a hyperthyroid cat, the overactive thyroid
revs up the kidneys and liver, too. So once the hyper-T
is corrected you may find that they do indeed have impaired
liver and kidney function--the first of which is dangerous for
long-term Tapazole use, the second of which is dangerous
for the radiation therapy.

Get me? If I got anything wrong, someone chime in.

Bottom line: I need to schedule the radiation soon, as her
levels of everything are fine. As for the expense--your kitty, and mine
are young enough that they could live another ten years. And
trust me, pilling twice a day sucks, even when you have a cat
that loves being handled.