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She was so absorbed in listening to the Beatles that she missed her date with him.
Literal
She [topic-は] Beatles [object-を] listen [nominalizer-の] [in-に] absorbed-was [and-て], him [with-の] date [to-に] missed-going.
Two patterns to watch here. のに (verb + nominalizer の + に) builds 'in/for the activity of doing X,' marking the focus or domain of attention: 聞くのに夢中になる ('be absorbed in listening'). Don't confuse this のに with the conjunction のに meaning 'although' — they look identical but parse differently. ~そびれる is an auxiliary verb attaching to the connective stem of another verb to mean 'miss the chance to do, fail to do' — 行きそびれる ('miss going'), 言いそびれる ('miss saying'), 食べそびれる ('miss eating'). It expresses a regretful missed opportunity. 彼との is a noun-modifying form: 彼と ('with him') + の ('of') = 'with-him's', i.e. 'the date with him.' The Beatles were huge in Japan — their 1966 Tokyo concerts were a cultural event whose echoes still shape how Japanese music fans relate to Western pop.