Some inspired suggestions (especially "Happy Birthday"). But I reckon that girl's skipping songs might be what you'd actually want - I imagine they might help keep the rapid breathing that's supposed to be useful in these situations. And the words fit well enough too in some cases,as well as being simple and repetitive, and comforting to people who grew up with them:

The big ship sailed down the alley-alley-o
the alley-alley-o
the alley-alley-o.
The big ship sailed down the alley-alley-o
on the last day of September

Or

In and out the dusty bluebells,
In and out the dusty bluebells,
In and out the dusty bluebells,
Who will be my neigbour.

And so on.