I agree with your thoughts on the dose.
As for the sharp rise upon waking up, this sounds like 'Foot on the floor', where your friendly liver dumps some glucose in your bloodstream to give you an energy boost for the day. The liver doesn't realise you're not dealing too well with sudden glucose dumps.
Some of us have found it helps to take a small dose of quick acting insulin before getting out of bed. To my thinking it doesn't matter if the glucose comes from my liver or from food, it needs insulin either way.
But be very careful, and don't mess with your doses on your own if you're not used to adjust your insulin! You can always contact your diabetes nurse to discuss this approach.
Do you take your Levemir once a day in the evening or do you split it?
If taken once a day, it can taper off in the hours before the next dose, leading to a rise.
It can also be your insulin to carbs ratio is a bit off in the afternoon of course.