Hi Tina,
Absolutely your vet is not dosing correctly, and dosing incredibly dangerously at that. Humulin is a very harsh insulin with steep drops, and raising your cat up to 8 units in the course of 3 days is super, super dangerous! Please do not shoot that much insulin tonight!
I have a few questions--did your vet mention if he tested Arwen for ketones, and if they were present? This can sometimes lead to slighting higher dosing (i.e. starting with 2u of insulin instead of 1u), but never 8u in 3 days. And you never raise the dose that quickly. Even if he didn't you should test for ketones regularly as uncontrolled diabetics are at risk for diabetic ketoacidosis, a serious and potentially deadly complication from diabetes. You can pick the urine test strips up at any drug store--they're called ketostix. If you get ketodiastix it tests for both ketones and glucose at the same time.
However, I would highly recommend you pick up a glucose meter and start blood testing the ear instead. Urine testing will not tell you if Arwen's blood sugar is going too low, and it's very important if you think there is a potential hypoglycemic situation to be able to see if her blood sugar is low so that you can treat it before it gets too bad. You can use any human meter and test strips. The Walmart Relion Micro is a good meter with really inexpensive test strips, but you can use any meter you prefer. Home blood testing before each shot is the only way to make sure it is safe to shoot insulin. You'll also need some lancets--26-28g work best. There are tons of tips and tricks here to help you get started if you need help! Home testing seems scary at first, but the large, large majority of cats do not mind it all within a week or two once they get used to it.
Humulin N is also not the recommended insulin for cats. It works well with dogs, but cats have faster metabolisms and need a long lasting insulin like Lantus or Levemir. I would urge you to get a prescription for one of these instead of the Humulin. Lantus and Levemir has a 84% remission rate in newly diagnosed cats. I'd be happy to provide the study to print and give to your vet so that he'll be on board with you changing insulins. Or you could switch vets to one a bit more knowledgeable about feline diabetes and start fresh.
What are you feeding Arwen? It's very important diabetic cats are feed low carb, canned diets. Dry food, even the prescription dry food, is high in carbs and will keep Arwen's blood sugar high. There are many commercial canned foods that are inexpensive and good to feed diabetics (for example, all the of the Fancy Feast "classic" flavors are diabetic safe). Many people are concerned about switching because they want to have food available when they are at work--but you can freeze portions of canned food to leave out or just leave some out with water added to it so it doesn't dry out. However, I would not recommend changing the diet until you change insulins, as Humulin is very harsh and prone to causing hypoglycemic incidents.
Please ask lots of questions! Diabetes is very treatable with the right plan, and my cat is healthier today than before he was diagnosed! It's overwhelming at first, but once you start blood testing and get her on a better insulin, a lot of this stress will subside.