You're traveling with a 2-year-old, who will need naps and such.

I'd suggest staying at an inexpensive place not far from the airport; there are many. You would end up with much less moving about, taking of trains, and crowds.

Just did a Google search. One place that came up is near "Wolmido Amusement Park, and Songwol-dong Fairy Tale Village". Might be perfect for your young child. It's $39.
https://www.expedia.com/Incheon-Hote...el-Information

I've never stayed there, but have stayed at nice, inexpensive places in the Incheon (airport) area.