She bit her lip with regret, wishing she hadn't said those things that got on her mother-in-law's nerves.

Literal

She [topic-は] mother-in-law [possessive-の] feeling [object-を] touching [like-ような] thing [object-を] saying not-having good-was [quotative-と] navel [object-を] bit.

姑 (しゅうとめ, 'mother-in-law') is the specific term. 気にさわる ('to get on someone's nerves') is the idiom. ~なければよかったと uses the regret pattern ('if only she hadn't'). 臍を噛む (ほぞをかむ, 'to bite one's navel') is a literary idiom meaning 'to regret bitterly' — physically impossible, hence the intensity of the metaphor. The sentence layers two idioms within a complex conditional-regret structure.