It sure sounds like you could be a type 1.With that low of carb diet and your daughter being a type 1that would make complete sense. You need to be the squeaky wheel. Too many type 1's find out the hard way by going into DKA. I was misdiagnosed for over 8 years but luckily I was put on insulin early because the meds made me sick and I refused to take them. I truly don't know if you are a type 1, but it sure seems like you could be.
I was not obese, but I was overweight so they just labeled me a type 2. I had an uncle that was a type 1. I ate really healthy, I swam 75 laps in a gym pool daily and it didn't make sense to me that I was a type 2. I was told no a couple of times when I asked by a specialist and my GP and they never even tested me. It wasn't until I switched doctors that I was tested and diagnosed right. Unfortunately 40% of type 1's being misdiagnosed is a current number, plus 50%of type 1's are diagnosed after the age of 30.
I think you need to stress your daughter's a type 1, and maybe how few carbs you are eating so you can get tested. Either that or switch doctors? I think it's more tricky in the UK so maybe someone has ideas how to go about getting tested for being a type 1.
The other thing I want to stress is getting ketone strips.They are pretty cheap here. About $6-10 a box. If you have any significant amount of ketones and you very well might since your blood sugar numbers have gone up so much you need to get to a hospital. DKA is caused from a lack of insulin, which type 1's are prone to when not getting insulin.
https://www.diabetes.org/diabetes/complications/dka-ketoacidosis-ketones