It took three days - three very long, excruciating days of being completely separated from the TARDIS before what lay dormant within Clara woke up. It had been in various states of awakening for some time - the little things: communicating with the TARDIS, healing herself, saying things out of time - sometimes right before something would happen. But, after three days of going through withdrawal, there was a fire-like glow to her - reminiscent of Bad Wolf, and yet, different. She was still Clara, just, Clara....plus, something.
She could see the events more clearly now - in more depth, more time, more layers. She had been in the timestream, yes. She had been torn to pieces, yes. But, where was the timestream itself? Inside her Doctor's dying TARDIS - the giant, leaking, dying TARDIS. That energy seeped into the cracks - she had already felt it as they climbed their way to the console room to start, it wasn't an unwelcome or unfamiliar presence when it found her in shatters.
When a doll is thrown into a storm - ripped and torn apart - does the storm put it back together? No. Thread with a bit of love does. That Old Girl gently wove the rest of her power into Clara, like delicate thread to piece her back together. How else would River know she was still alive? Mentally linked through the TARDIS? Perhaps.
If she had to use a term to describe herself, the closest term she could think of is a TARDIS-Human-Meta-Crisis. Part TARDIS/Part Human. She won't live forever - but she has a significantly longer lifespan than the humans she knows and certainly isn't as fragile, the Old Girl had life to give. She is very aware of where she is in time - not in the same sense of a Time Lord, just an awareness with a strong pull toward certain events now and again.
And Chin-Boy's TARDIS? ...they got along from the start because perhaps they felt a kindred spirit in each other. For his Old Girl, Clara would feel the closest to home she's known in a long time.
The glow fades, and Clara finally has the answers she's been searching for since she landed.