I started saying Modeh Ani with my daughter every morning which is great. She's a baby and she takes a few naps a day.
Should I say it Every time she wakes up or just in the morning?
I don't know for sure, but I am fairly sure we say Modeh Ani whenever we awake, not just in the morning. And we always wash our hands. At least that's how I was taught.
We should always thank Hashem whenever he restores our soul.
As far as I know, the minhag is only to say modeh ani upon waking in the morning, not after subsequent naps.