Waking up at 5.2 is fine. What targets did they give you? I'd stay with the dose you are using for a few days & let everything settle a bit before altering again. It may not be the lantus that is dropping you from 10 to 5.2 but the residual novorapid from your evening meal. If your meal was high in protein this would take the novorapid longer to deal with.
When I started, I devised a boring but repeatable diet & pretty much ate the same things for the first few weeks. This allowed me to find the right Novorapid for the meals I was using. Once I had that under control I turned to sorting out the Lantus.
I had been started on 20units by the hospital clinic, who explained that I should alter it up or down using the morning blood tests. If low, decrease the dose if high increase. However they also pointed out that I shouldn't alter more than 2 units either way & allow 2 to 3 days for the changes to work through.
They also explained that the 'honeymoon' period, when the pancreas starts to provide more insulin as your injected insulin takes some load from it & you recover from the initial symptoms that got you to the doctor in the first place.
Once you've settled down a bit, you could do some basal testing overnight, which is a pain as you have to set your alarm & check your blood every hour. Do a search on this forum & you'll find plenty of how-to's for it.
Hope this helps.