First, have your sister join this group, Hyperthyroid Cats on Facebook.
https://www.facebook.com/groups/42301610484 If she isn't on Facebook, it's worth joining just for the info this group has. Really.
After two years being treated with medicine, and our cat's thyroid levels still rising, we had the iodine treatment done.
The time before treatment when he had to be off his medicine was very difficult. His levels were sky high, and the poor cat was absolutely miserable with extreme hyperthyroid symptoms. But this is a necessary evil in order for treatment.
The treatment itself is easy--they give the cat a measured injection of the I131. The cat then has to stay at the facility until the level of radioactivity decreases below a set limit. This will depend on your state/country. Ours lets most cats go in 3 days. I know of other countries where they hold the cat for 2-3 weeks, until all radioactivity is gone.
You cannot visit your cat during this time. Each cat will be in his own little "cat condo" (as they usually call it). You can send your own blanket or other stuff with them, but you will not get it back (because it will be exposed to radioactivity.) We had hubby sleep on top of an old t-shirt for a week, so that it smelled like him, and sent that along.
When your cat comes home, he will still have a low level of radioactivity. So the instructions will be to avoid close, frequent, or prolonged contact with the cat. So no snuggling on your lap or sleeping with you, don't hold and carry them around, etc. Best if they don't sleep with other pets, too. This is because you/the other pets will then be being exposed to the low level of radiation. As people live many many years longer than cats, this is something they want you to consider, as any effects from your exposure will have decades to show up. If your cat's buddies are also old cats, probably wouldn't be a factor as they will pass long before a problem can develop. Many people find this "isolation" protocol difficult, and ignore it. Since we had very young cats also, 2-3 year olds who could live another 20 years, we chose to pay for a few extra days at the vet's to make sure our cat's levels were much much lower before bringing him home. It was tough to be away from him, but the place would call with daily updates, and we felt the increased safety for the youngsters and us was worth it.
You can still play with your cat and all, just best to do it with things like string on a stick and laser light pointers, stuff that keeps you a few feet apart, as that is all that is necessary to avoid any residual radiation.
His pee and poop will contain radioactivity, again, a very low level, so they instruct you to collect it all, and usually say hold it for 30 days (or up to three months, depending on where you are) then you can throw it out. By that time the radiation has disintegrated, and it's safe. You can do this by just having a small, covered, trash pail outside your house, or if in an apartment, in the farthest, most unused corner of the place or back of a closet. This is not powerful radioactivity, so keeping a few feet away or a wall or two between you and it is sufficient.
We were not instructed to throw out bowls or bedding or anything. If your cat sleeps 20 hours a day on the same cat bed, you might want to put it in isolation for a while after the cat is deemed "safe", just in case, but probably not necessary.
Occasionally, a cat goes HYPOthyroid after treatment, ours did, but that can be controlled very easily with thyroid medicine pills. Our cat gets his tiny pill twice a day, with breakfast and dinner, no problem. Inexpensive (have them give you a prescription to take to a human pharmacy, so much cheaper!). Will need occasional bloodwork to make sure he's still at a good level and that medicine doesn't need adjusting. No problem.
It IS very expensive, but when you consider the cost of treating hyperthyroid when you keep having to up the medication dose, it doesn't take long to even out the costs. And the I131 is a CURE. I wish we had just done it from the start.