Try looking into The Dawn Phenomenon. For reasons not entirely understood, blood sugars tend to rise for 2-3 hours after waking. One theory is that some bodily organs shut down a bit during sleep, and when we awake the body overcompensates, releasing glycogen stores which cause blood sugars to rise. The other things to consider are your injection sites, your insulin pen and your reusing of needles. To keep insulin absorption good, it's really important to rotate your injection sites. Until a few weeks ago I did not pay this much heed. My DSN took one look at my stomach from across the room and told me exactly where I injected all the time, because a lump was fairly obvious. I started avoiding that area, and suddenly my blood sugars were through the floor. I was amazed how much less insulin I required. It's also best to check your insulin pen and make sure it is working correctly. Every now and then I squirt a couple of units into the sink before I inject, in order to see that insulin is coming out. Lastly, insulin absorption will suffer by reusing needles. Again, I did not worry too much about this, but my control was suffering. Changing needles after every single injection has helped make my insulin requirements more predictable.
If all else fails, it may be worth speaking to your diabetes team about changing insulin type. Levemir is not the only brand on the market, and another type (such as humalog) may suit you better. Best of luck!