Does this video catch an Eta Aquarid - Earth skimming meteor? I did a time-lapse animation looking towards the sunrise from Sydney, Australia on 25th of April.  You can see it here around 1m 25s in (that link should jump to just before it appears very near where the Sun is about to rise):

http://www.youtube.com/watch?v=lqr_KuLd2-c&amp;t=1m25s

It then proceeds towards the right of screen over around the next 21 seconds (around 10&#189; minutes) before disappearing off the right. You might need to send the video to full screen to see the small white 'bullet of haze' that is moving against a very wide background.

One of my first thoughts was that it might be a meteor skimming the upper atmosphere, but I was less confident of that given it seemed to be coming from the general direction of the Sun, and I'd thought that was a direction unlikely to produce meteors. 

That was until I read Eta Aquarid Meteor Shower in 2016 which states:


  The best time to view the Eta Aquarids is in the early mornings, right before dawn.


Well, at least that pins down the same time of day, if not the direction from which it came.

What is the likelyhood that this feature is a piece of comet debris that skimmed Earth's atmosphere?
Things entering the atmosphere (whether a meteor or an atmosphere-skimming object) travel too fast to be visible for 10 minutes.  Normally a pass of a satellite (at above atmosphere height) takes 2-3 minutes and most meteors are over in seconds.

Even the famous daylight fireball from the 1970s was over in a very short time.

I would guess what you have there is a plane at high altitude, though it's moving a bit slower than I would expect. (I'm open to suggestions.)