Eagle Auto Router Display
The line you mention means that there are 659 connections to be made and 635 of them ended up made. Also, the autorouter placed 605 vias (which seems like a lot to me).  

You should check the DRC, specifically the clearance tab.  You might have the clearances set looser than they need to be for the fabricator you're using.  For example, maybe you could move traces nearer to pads than you are right now.

If the routing reaches 100% and then drops back, I find that that's usually because the ground plane fell apart, and now some things that seemed grounded have lost their reference.  Paradoxically, the way I can often fix that is to set my DRC to leave MORE room around my vias, so that the ground plane can sneak through them.  This may do bad things to my copper pour, so I manually check to make sure that nothing bad happened where I most need continuous ground planes.

You need to find out where your failures to route are.  Hit ratsnest, and view layer 19 in isolation if the misses aren't apparent.  Look for the signals on the unrouted traces.  If they're mostly GND, your problem is the ground plane breaking up.

If all else fails, consider more layers, or routing the tough bits by hand.