She refused to play with it and just sat there pouting.

Literal

She [topic-は] it [with-で] play [nominalizer-こと] [object-を] refused [then-て], just sit [then-て] was-pouting [explanatory-のです].

拒む is a stronger, more deliberate refusal than 断る — there is a stubborn, willful weight to it, as if planting one's heels. Pairing it with the こと nominalizer (rather than の) suits this firmer, more abstract framing: 'the act of playing.' ふくれる literally means 'to swell' or 'inflate,' but figuratively describes the cheek-puffing pout of a sulking child. The closing ~のです gives the sentence a narrative or explanatory texture, like setting a scene for someone listening. Together it paints a small tantrum without raising the tone of voice.