Well Ruth it's bitter sweet to see you here; on the one hand if having cancer was not a big enough challenge it appears you're treatment has caused diabetes, and in the other like
@Pipp said I'm sure people will be only too pleased to help.
The newly diagnosed tend to have a so called honeymoon period where beta cells may still produce insulin. You can get a test for C-Peptide (a protein that binds to insulin I believe) which is indicative of insulin production.
It sounds like you have been given set amounts of Levemir and separately set amounts of fast acting and it would be interesting to know the process by which these were prescribed.
One thing I'd hope is that you should have priority access to both cancer and diabetes specialist nurses especially as I have heard - only through rudimentary research - high glucose and separately high stress is generally a poor environment to reduce cancer cell multiplication and that your diabetes may be affected in some way of which we are unaware by your immunotherapy.
If you haven't already I would really recommend getting as much time as you can with your oncologist, endocrinologist and aforementioned nurses to focus on well being in the context of both diagnoses.
On the sudden lows this, as you've probably found out can be really dangerous.
If this next bit sounds condescending then please forgive me but I'm assuming you have not been advised how to count carbohydrates. If you are consistently giving 6 units at a meal but are having variable meals then this may cause problems. In the extreme if you give 6 units and your glucose level comes back to the level it started before you ate say, a baked potato with salad and then a banana, but the next day you gave the same injection but only eat green salad (no carbohydrate) your glucose levels are likely to plummet. If you're taking a fixed dose then fixed amounts carbohydrates nay help.
Generally I imagine your dietician has advocated eating real food with lots of vegetables rather than ultra processed products at the other extreme but if not I'd suggest asking about that in general terms but would expect that to be of benefit to most people.
It would probably be helpful to the community to understand how you have to manage your lifestyle, specifically insulin dosing regimen and dietary advice you've received with these two serious diseases.
I do hope this helps but will look in on the thread later as I'm sure you'll have lots else to read.
All the very best
Mat