If I've understood you correctly, you are getting a spike after your breakfast but your bg before lunch is okay?
Sid's right about reducing carbs if you were on standard T2 treatments, but you're on insulin now and the good thing about that is that you can eat all the carbs you want as long as you cover it with the correct injection dose- and from your lunch readings I'd say that you've got the carb/insulin ratio right. (are you confident altering your insulin dose to match your carbs?)
three options I would consider:
1) if you are injecting after your meal, or injecting with your meal, try injecting even earlier. Eg inject 10-15 mins before you eat breakfast. This will give the novorapid a chance to be kicking in at the same time as your sugar spike kicks in.
2) look for slower-absorbed carbs, eg try porridge or wholemeal toast rather than rice krispies or white toast (I don't know what you are currently eating so this may not help).
3) choose not to worry about it at all. Balancing insulin is a difficult job and often you can find yourself 'chasing' good bg readings, sending yourself hypo between meals, as you've found out. If you don't feel unwell with the bg spike after breakfast, and your lunchtime readings stay reasonable, then you should be fine long-term, to be honest. Obviously, if your next HbA1c is high you'll need to rethink, but I would bet you'll be okay. Current (DAFNE) thinking for those on insulin is not to test between mealtimes at all, to avoid swinging blood sugars and chasing readings. After more than two decades of self-managing insulin, I'd say that was good advice.