Seattle to Taiwan to Beijing to Taiwan to Seattle, does that work for the transit visa?

My sister is coming to see me in Taiwan. She's flying from Seattle to Beijing but has a long layover in Taiwan first where I'll get to see her and then join her in the flight over to Beijing. Afterwards we are both flying back to Taiwan for a while, then she goes back to Seattle. Does this work for the transit? The original flight is to Beijing, but just has a layover in Taiwan. Not sure if this would mess up the transit type. Anyone know? Thanks!